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--- abstract: 'Several results concerning existence of solutions of a quasiequilibrium problem defined on a finite dimensional space are established. The proof of the first result is based on a Michael selection theorem for lower semicontinuous set-valued maps which holds in finite dimensional spaces. Furthermore this result allows one to locate the position of a solution. Sufficient conditions, which are easier to verify, may be obtained by imposing restrictions either on the domain or on the bifunction. These facts make it possible to yield various existence results which reduce to the well known Ky Fan minimax inequality when the constraint map is constant and the quasiequilibrium problem coincides with an equilibrium problem. Lastly, a comparison with other results from the literature is discussed.' author: - Marco Castellani - Massimiliano Giuli - Massimo Pappalardo title: A Ky Fan minimax inequality for quasiequilibria on finite dimensional spaces --- Introduction ============ In [@Fa72] the author established the famous Ky Fan minimax inequality which concerns the existence of solutions for an inequality of minimax type that nowadays is called in literature “equilibrium problem”. Such a model has gained a lot interest in the last decades because it has been used in different contexts as economics, engineering, physics, chemistry and so on (see [@BiCaPaPa13] for a recent survey). In these equilibrium problems the constraint set is fixed and hence the model can not be used in many cases where the constraints depend on the current analyzed point. This more general setting was studied for the first time in the context of impulse control problem [@BeGoLi73] and it has been subsequently used by several authors for describing a lot of problems that arise in different fields: equilibrium problem in mechanics, Nash equilibrium problems, equilibria in economics, network equilibrium problems and so on. This general format, commonly called “quasiequilibrium problem”, received an increasing interest in the last years because many theoretical results developed for one of the abovementioned models can be often extended to the others through the unifying language provided by this common format. Unlike the equilibrium problems which have an extensive literature on results concerning existence of solutions, the study of quasiequilibrium problems to date is at the beginning even if the first seminal work in this area was in the seventies [@Mo76]. After that, the problem concerning existence of solutions has been developed in some papers [@AlRa16; @Au93; @AuCoIu17; @CaGi15; @CaGi16; @Cu95; @Cu97]. Most of the results require either monotonicity assumptions on the equilibrium bifunction or upper semicontinuity of the set-valued map which describes the constraint. Whereas other authors provided existence of solutions avoiding any monotonicity assumption and assuming lower semicontinuity of the constraint map and closedness of the set of its fixed points. Aim of this paper is to establish several results concerning existence of solutions of a quasiequilibrium problem defined on a finite dimensional space which comes down to the Ky Fan minimax inequality in the classical setting. Our approach is based on a Michael selection result [@Mi56] for lower semicontinuous set-valued maps. Moreover the proof of our results allow one to locate the position of a solution. The paper is organized as follows. Section 2 is devoted to recall the results about set-valued maps which are used later. In Section 3 we prove the main theorem and we furnish more tractable conditions on the equilibrium bifunction which guarantee that our result holds true. Basic concepts ============== Let $\Phi:X\rightrightarrows Y$ be a set-valued map with $X$ and $Y$ two topological spaces. The graph of $\Phi$ is the set $${\operatorname{gph}}\Phi:=\{(x,y)\in X\times Y:y\in \Phi(x)\}$$ and the lower section of $\Phi$ at $y\in Y$ is $$\Phi^{-1}(y):=\{x\in X:y\in \Phi(x)\}.$$ The map $\Phi$ is said to be lower semicontinuous at $x$ if for each open set $\Omega$ such that $\Phi(x)\cap\Omega\ne\emptyset$ there exists a neighborhood $U_x$ of $x$ such that $$\Phi(x')\cap\Omega\ne\emptyset,\qquad\forall x'\in U_x.$$ Notice that a set-valued map with open graph has open lower sections and, in turn, if it has open lower sections then it is lower semicontinuous. A fixed point of a function $\varphi:X\rightarrow X$ is a point $x\in X$ satisfying $\varphi(x)=x$. A fixed point of a set-valued map $\Phi:X\rightrightarrows X$ is a point $x\in X$ satisfying $x\in\Phi(x)$. The set of the fixed point of $\Phi$ is denoted by ${\operatorname{fix}}\Phi$. One of the most famous fixed point theorems for continuous functions was proven by Brouwer and it has been used across numerous fields of mathematics (see [@Bo85]). .3truecm [**Brouwer fixed point Theorem.**]{} * Every continuous function $\varphi$ from a nonempty convex compact subset $C\subseteq{{\mathbb R}}^n$ to $C$ itself has a fixed point.* .3truecm A selection of a set-valued map $\Phi:X\rightrightarrows Y$ is a function $\varphi:X\rightarrow Y$ that satisfies $\varphi(x)\in\Phi(x)$ for each $x\in X$. The Axiom of Choice guarantees that set-valued maps with nonempty values always admit selections, but they may have no additional useful properties. Michael [@Mi56] proved a series of theorems on the existence of continuous selections that assume the condition of lower semicontinuity of set-valued maps. We present here only one result [@Mi56 Theorem 3.1$^{\prime\prime\prime}$ (b)]. .3truecm [**Michael selection Theorem.**]{} * Every lower semicontinuous set-valued map $\Phi$ from a metric space to ${{\mathbb R}}^n$ with nonempty convex values admits a continuous selection.* .3truecm The Michael selection Theorem holds more in general when the domain of $\Phi$ is a perfectly normal space. Collecting the Brouwer fixed point Theorem and the Michael selection Theorem, we deduce the following fixed point result for lower semicontinuous set-valued maps. \[cor:fixed point\] Every lower semicontinuous set-valued map $\Phi$ from a nonempty convex compact subset $C\subseteq{{\mathbb R}}^n$ to $C$ itself with nonempty convex values has a fixed point. Notice that, unlike the famous Kakutani fixed point Theorem (see [@Bo85]) in which the closedness of ${\operatorname{gph}}\Phi$ is required, in Corollary \[cor:fixed point\] the lower semicontinuity of the set-valued map is needed. No relation exists between the two results as the following example shows. The set-valued map $\Phi:[0,3]\rightrightarrows [0,3]$ $$\Phi(x):=\left\{\begin{array}{ll} \{1\} & \mbox{ if } 0\leq x\leq 1\\ (1,2) & \mbox{ if } 1<x<2\\ \{2\} & \mbox{ if } 2\leq x\leq 3 \end{array}\right.$$ is lower semicontinuous and the nonemptiness of ${\operatorname{fix}}\Phi$ is guaranteed by Corollary \[cor:fixed point\]. Notice that ${\operatorname{fix}}\Phi=[1,2]$. Nevertheless the Kakutani fixed point Theorem does not apply since ${\operatorname{gph}}\Phi$ is not closed. On the converse, the set-valued map $\Phi:[0,3]\rightrightarrows [0,3]$ $$\Phi(x):=\left\{\begin{array}{ll} \{1\} & \mbox{ if } 0\leq x<1\\ {}[1,2] & \mbox{ if } 1\leq x\leq 2\\ \{2\} & \mbox{ if } 2<x\leq 3 \end{array}\right.$$ has closed graph and the nonemptiness of ${\operatorname{fix}}\Phi$ is guaranteed by the Kakutani fixed point Theorem. Again ${\operatorname{fix}}\Phi=[1,2]$. Since $\Phi$ is not lower semicontinuous, Corollary \[cor:fixed point\] can not be applied. We conclude this section recalling some topological notations. Given two subsets $A\subseteq C\subseteq{{\mathbb R}}^n$ we denote by ${\operatorname{int}}_C A$ and ${\operatorname{cl}}_C A$ the interior and the closure of $A$ in the relative topology of $C$ while $\partial_C A$ indicates the boundary of $A$ in $C$, i.e. $$\partial_C A:={\operatorname{cl}}_C A\setminus {\operatorname{int}}_CA = {\operatorname{cl}}_C A\cap {\operatorname{cl}}_C (C\setminus A).$$ Lastly $C$ is connected if and only if the subsets of $C$ which are both open and closed in $C$ are $C$ itself and the empty set. Existence results ================= From now on, $C\subseteq {{\mathbb R}}^n$ is a nonempty convex compact set and $f:C\times C\rightarrow {{\mathbb R}}$ is an equilibrium bifunction, that is $f(x,x)=0$ for all $x\in C$. The equilibrium problem is defined as follows: $$\label{eq:ep} \mbox{find } x\in C \mbox{ such that } f(x,y)\ge 0\mbox{ for all } y\in C.$$ Equilibrium problem has been traditionally studied assuming that $f$ is upper semicontinuous in its first argument and quasiconvex in its second one. Under such assumptions, the issue of sufficient conditions for existence of solutions of (\[eq:ep\]) was the starting point in the study of the problem. Ky Fan [@Fa72] proved a famous minimax inequality assuming compactness of $C$ and his result holds in a Hausdorff topological vector space. However, there is the possibility to slightly relax the continuity condition when the vector space is finite dimensional. The set-valued map $$\label{eq:mapF} F(x):=\{y\in C:f(x,y)<0\}$$ defined on $C$ plays a fundamental role in the formulation of our results. Clearly $F$ has open lower sections and convex values under the Ky Fan assumptions on the bifunction $f$, that is upper semicontinuity with respect to the first variable and quasiconvexity with respect to the second one. The fact that $F$ has open lower sections implies that $F$ is lower semicontinuous. If $F$ had nonempty values, Corollary \[cor:fixed point\] guarantees the existence of a fixed point of $F$. This contradicts the fact that $f(x,x)\geq 0$. Therefore there exists at least one $\bar x$ such that $F(\bar x)=\emptyset$, that is a solution of the equilibrium problem (\[eq:ep\]). The following result holds. .3truecm [**Ky Fan minimax inequality.**]{} *A solution of (\[eq:ep\]) exists whenever the set-valued map $F$ given in (\[eq:mapF\]) is lower semicontinuous and convex-valued.* .3truecm After describing this auxiliary result, we focus on the main aim of the paper. A quasiequilibrium problem is an equilibrium problem in which the constraint set is subject to modifications depending on the considered point. This format reads $$\label{eq:qep} \mbox{find } x\in K(x) \mbox{ such that } f(x,y)\ge 0\mbox{ for all } y\in K(x),$$ where $K:C\rightrightarrows C$ is a set-valued map. Our first existence result is the following. \[th:existenceQEP\] Assume that $K$ is lower semicontinuous with nonempty convex values and ${\operatorname{fix}}K$ is closed. Moreover suppose that 1. $F$ is convex-valued on ${\operatorname{fix}}K$, 2. $F$ is lower semicontinuous on ${\operatorname{fix}}K$, 3. $F\cap K$ is lower semicontinuous on $\partial_C{\operatorname{fix}}K$, where $F$ is the set-valued map given in (\[eq:mapF\]). Then the quasiequilibrium problem (\[eq:qep\]) has a solution. [**Proof.**]{} Corollary \[cor:fixed point\] ensures the nonemptiness of ${\operatorname{fix}}K$. If ${\operatorname{fix}}K=C$, the existence of solutions to the quasiequilibrium problem descends from the above mentioned Ky Fan minimax inequality. Otherwise, since ${\operatorname{fix}}K$ is closed and $\partial_C{\operatorname{fix}}K ={\operatorname{fix}}K\setminus {\operatorname{int}}_C{\operatorname{fix}}K$, the emptiness of $\partial_C{\operatorname{fix}}K$ it would be equivalent to ${\operatorname{fix}}K={\operatorname{int}}_C{\operatorname{fix}}K$. Therefore ${\operatorname{fix}}K$ would be both open and closed in $C$. Since every convex set is connected, the only nonempty open and closed subset of $C$ is $C$ itself and this contradicts the fact that ${\operatorname{fix}}K\ne C$. Assume that ${\operatorname{int}}_C{\operatorname{fix}}K\ne\emptyset$ (the case ${\operatorname{int}}_C{\operatorname{fix}}K=\emptyset$ is similar and will be shortly discussed at the end of the proof) and define $G:C\rightrightarrows C$ as follows $$G(x):=\left\{\begin{array}{ll} F(x) & \mbox{ if } x\in {\operatorname{int}}_C{\operatorname{fix}}K\\ F(x)\cap K(x) & \mbox{ if } x\in \partial_C{\operatorname{fix}}K\\ K(x) & \mbox{ if } x\notin {\operatorname{fix}}K \end{array}\right.$$ The proof is complete if we can show that $G(x)=\emptyset$ for some $x\in C$. Indeed, since $K$ has nonempty values, then $x\in{\operatorname{fix}}K$ and two cases are possible. If $x\in\partial_C{\operatorname{fix}}K$, then it solves (\[eq:qep\]); if $x\in{\operatorname{int}}_C{\operatorname{fix}}K$ then it solves (\[eq:ep\]). In both cases the quasiequilibrium problem has a solution. Assume by contradiction that $G$ has nonempty values. Next step is to prove the lower semicontinuity of $G$. Fix $x\in C$ and an open set $\Omega\subseteq{{\mathbb R}}^n$ such that $G(x)\cap\Omega\cap C\ne\emptyset$. We distinguish three cases. 1. If $x\in{\operatorname{int}}_C{\operatorname{fix}}K$, from the lower semicontinuity of $F$ there exists a neighborhood $U'_x$ such that $$F(x')\cap\Omega\cap C\ne\emptyset,\qquad\forall x'\in U'_x\cap{\operatorname{fix}}K$$ which implies $$G(x')\cap\Omega\cap C\ne\emptyset,\qquad\forall x'\in U'_x\cap{\operatorname{int}}_C{\operatorname{fix}}K.$$ Since $U'_x\cap{\operatorname{int}}_C{\operatorname{fix}}K$ is open in $C$, then $G$ is lower semicontinuous at $x$. 2. If $x\in\partial_C{\operatorname{fix}}K=\partial_C(C\setminus{\operatorname{fix}}K)$ from the lower semicontinuity of $F$, $K$ and $F\cap K$ there exist neighborhoods $U'_x$, $U''_x$ and $U'''_x$ such that $$\begin{aligned} F(x')\cap\Omega\cap C\ne\emptyset, & \qquad & \forall x'\in U'_x\cap{\operatorname{fix}}K,\\ K(x')\cap\Omega\cap C\ne\emptyset, & \qquad & \forall x'\in U''_x\cap C,\\ F(x')\cap K(x')\cap\Omega\cap C\ne\emptyset, & \qquad & \forall x'\in U'''_x\cap \partial_C{\operatorname{fix}}K.\end{aligned}$$ Then $$G(x')\cap\Omega\cap C\ne\emptyset,\qquad\forall x'\in U'_x\cap U''_x\cap U'''_x\cap C,$$ i.e. $G$ is lower semicontinuous at $x$. 3. Finally, if $x\notin{\operatorname{fix}}K$, from the lower semicontinuity of $K$ there exists a neighborhood $U'_x$ such that $$K(x')\cap\Omega\cap C\ne\emptyset,\qquad\forall x'\in U'_x\cap C.$$ Then $$G(x')\cap\Omega\cap C\ne\emptyset,\qquad\forall x'\in U'_x\cap (C\setminus{\operatorname{fix}}K).$$ Since $U'_x\cap(C\setminus{\operatorname{fix}}K)$ is open in $C$, then $G$ is lower semicontinuous at $x$. Since by assumption $G$ is also convex-valued, then all the conditions of Corollary \[cor:fixed point\] are satisfied and there exists $x\in{\operatorname{fix}}G$. Clearly $x\in{\operatorname{fix}}K$ and therefore $x\in{\operatorname{fix}}F$ which implies $f(x,x)<0$ and contradicts the assumption on $f$. The issue of ${\operatorname{int}}_C{\operatorname{fix}}K=\emptyset$ remains to be seen. In this case $\partial_C{\operatorname{fix}}K={\operatorname{cl}}_C{\operatorname{fix}}K={\operatorname{fix}}K$ and $G$ assumes the following form $$G(x):=\left\{\begin{array}{ll} F(x)\cap K(x) & \mbox{ if } x\in{\operatorname{fix}}K\\ K(x) & \mbox{ if } x\notin {\operatorname{fix}}K \end{array}\right.$$ The result is obtained by adapting the argument used before. It is clear from the proof that the assertion remains valid if $f(x,x)=0$ on $C\times C$ is replaced by the weaker $f(x,x)\ge0$ for all $x\in{\operatorname{fix}}K$. \[re:alternative\] The proof of Theorem \[th:existenceQEP\] allows to establish that a solution of (\[eq:qep\]) belongs to $$\partial_C{\operatorname{fix}}K\cup \{x\in {\operatorname{int}}_C{\operatorname{fix}}K: x \mbox{ solves } (\ref{eq:ep})\}.$$ In particular if (\[eq:ep\]) has no solution then Theorem \[th:existenceQEP\] ensures that a solution of (\[eq:qep\]) lies on the boundary of ${\operatorname{fix}}K$. \[re:fan\] By specializing to $K(x):=C$, for all $x\in C$, Theorem \[th:existenceQEP\] becomes the Ky Fan minimax inequality. Indeed ${\operatorname{fix}}K=C$ and conditions i) and ii) coincide with the assumptions in Ky Fan minimax inequality. Instead, since $\partial_C{\operatorname{fix}}K=\emptyset$, condition iii) is trivially satisfied. Theorem \[th:existenceQEP\] is strongly related to [@Cu95 Lemma 3.1]. The two sets of conditions differ only in that the lower semicontinuity of $F\cap K$ on the whole space $C$ assumed in [@Cu95 Lemma 3.1] is here replaced by the lower semicontinuity of $F$ on ${\operatorname{fix}}K$ and the lower semicontinuity of $F\cap K$ on $\partial_C {\operatorname{fix}}K$. We provide an example in which the results are not comparable to each other. Let $C:=[0,1]$ and $$f(x,y):=\left\{\begin{array}{ll} -1 & \mbox{ if } x=0 \mbox{ and } y\in(0,1]\\ 0 & \mbox{ otherwise} \end{array}\right.$$ If $K(x):=\{x\}$, for all $x\in [0,1]$, then $F\cap K=\emptyset$ is trivially lower semicontinuous and the assumptions of [@Cu95 Lemma 3.1] are satisfied. Instead $F$ is not lower semicontinuous at $0\in{\operatorname{fix}}K=[0,1]$. On the other hand if $K(x):=\{1-x\}$, for all $x\in [0,1]$, then ${\operatorname{fix}}K=\{1/2\}$, the assumptions of Theorem \[th:existenceQEP\] are trivially satisfied, but $F\cap K$ is not lower semicontinuous at $0$. It would be desirable to find more tractable conditions on $f$, disjoint from the ones assumed on $K$, which guarantee that all the assumptions i), ii) and iii) of Theorem \[th:existenceQEP\] are satisfied. Clearly the convexity of $F(x)$ can be deduced from the quasiconvexity of $f(x,\cdot)$ for all $x\in{\operatorname{fix}}K$. While the upper semicontinuity of $f(\cdot,y)$ on ${\operatorname{fix}}K$ implies that $F^{-1}(y)$ is open on ${\operatorname{fix}}K$ and hence $F$ is lower semicontinuous on ${\operatorname{fix}}K$. The last part of this section is devoted to furnish sufficient conditions for assumption iii), i.e. which guarantee the lower semicontinuity of the set-valued map $F\cap K$ on $\partial_C{\operatorname{fix}}K$. We propose two approaches. The former one consists in exploiting the following result in [@Pa91]. \[pr:lsc intersection\] Let $\Phi_1,\Phi_2:X\rightrightarrows Y$ be set-valued maps between two topological spaces. Assume that ${\operatorname{gph}}\Phi_1$ is open on $X\times Y$ and $\Phi_2$ is lower semicontinuous. Then $\Phi_1\cap\Phi_2$ is lower semicontinuous. Since $K$ is assumed to be lower semicontinuous, we investigate which assumptions ensure the open graph of $F$ given in (\[eq:mapF\]), that is the openness of the set $$\label{eq:open} \{(x,y)\in \partial_C{\operatorname{fix}}K\times C:f(x,y)< 0\}.$$ Hence, Theorem \[th:existenceQEP\] still works by using this condition instead of iii). It is interesting to compare this fact with [@Cu97 Theorem 2.1] where the openness of the set $\{(x,y)\in C\times C:f(x,y)< 0\}$ is required instead of the openness of (\[eq:open\]) and the lower semicontinuity of $F$ on ${\operatorname{fix}}K$. One should not overlook the fact that even though the results are formally similarly formulated, unlike our result, [@Cu97 Theorem 2.1] does not reduce to Ky Fan minimax inequality when $K(x)=C$, for all $x\in C$. An open graph result is [@Zh95 Proposition 2] which affirms that if $X$ is a topological space and $\Phi:X\rightrightarrows {{\mathbb R}}^n$ is a set-valued map with convex values, then $\Phi$ has open graph in $X\times{{\mathbb R}}^n$ if and only if $\Phi$ is lower semicontinuous and open valued. This fact has been used to establish the existence of continuous selections, maximal elements, and fixed points of correspondences in various economic applications. Up to translations, this result also holds when the codomain of $\Phi$ is an affine subset of ${{\mathbb R}}^n$ [@Yu98 Theorem 1.12]. We recall that an affine set of ${{\mathbb R}}^n$ is the translation of a vector subspace. Moreover, the affine hull of a set $C$ in ${{\mathbb R}}^n$, which is denoted by ${\operatorname{aff}}C$, is the smallest affine set containing $C$, or equivalently, the intersection of all affine sets containing $C$. \[th:sufficientconditions1\] Let $A\supseteq C$ be an open set on ${\operatorname{aff}}C$ and $\hat{f}:C\times A\rightarrow {{\mathbb R}}$ be a bifunction such that $\hat{f}(x,y)=f(x,y)$ for all $(x,y)\in C\times C$. Denote by $\hat{F}$ the set-valued map $$\hat{F}(x):=\{y\in A:\hat{f}(x,y)<0\}$$ defined on $C$ and assume that $K$ is lower semicontinuous with nonempty convex values and ${\operatorname{fix}}K$ is closed. Moreover suppose that 1. $\hat{F}$ is convex-valued on ${\operatorname{fix}}K$, 2. $\hat{F}$ has open lower sections on ${\operatorname{fix}}K$, 3. $\hat{F}(x)$ is open on ${\operatorname{aff}}C$ for all $x\in\partial_C{\operatorname{fix}}K$. Then the quasiequilibrium problem (\[eq:qep\]) has a solution. [**Proof.**]{} We have to show that all the assumptions of Theorem \[th:existenceQEP\] are fulfilled. Since the set-valued map $F$ given in (\[eq:mapF\]) can be expressed as $\hat F\cap C$, i) implies that $F$ is convex-valued on ${\operatorname{fix}}K$ and ii) implies that $F$ is open lower section on ${\operatorname{fix}}K$. In particular $F$ is lower semicontinuos on ${\operatorname{fix}}K$. Furthermore assumption iii) allows to apply [@Yu98 Theorem 1.12] which ensures that ${\operatorname{gph}}\hat F$ is open on $\partial_C{\operatorname{fix}}K\times {\operatorname{aff}}C$. Hence ${\operatorname{gph}}F={\operatorname{gph}}\hat F\cap (\partial_C{\operatorname{fix}}K\times C)$ is open on $\partial_C{\operatorname{fix}}K\times C$ and Proposition \[pr:lsc intersection\] guarantees that the intersection map $F\cap K$ is lower semicontinuous on $\partial_C{\operatorname{fix}}K$. The open graph result [@Zh95 Proposition 2] no longer holds when ${{\mathbb R}}^n$ (or an affine space) is replaced with an infinite dimensional Hilbert space [@Ba12]. However if $C\subset{{\mathbb R}}^n$ is a polytope, that is the convex hull of a finite set, then every $\Phi:X\rightrightarrows C$ with open lower sections and convex open values has open graph [@Bo85 Proposition 11.14]. This fact can be used for proving our next result. \[th:sufficientconditions2\] Assume that $C$ is a polytope and $K$ is lower semicontinuous with nonempty convex values and ${\operatorname{fix}}K$ is closed. Moreover suppose that 1. $F$ is convex-valued on ${\operatorname{fix}}K$, 2. $F$ has open lower sections on ${\operatorname{fix}}K$, 3. ì$F(x)$ is open on $C$ for all $x\in \partial_C {\operatorname{fix}}K$, where $F$ is the set-valued map given in (\[eq:mapF\]). Then the quasiequilibrium problem (\[eq:qep\]) has a solution. [**Proof.**]{} The set-valued map $F$ has open lower sections, convex and open values. Then its graph is open on $\partial_C {\operatorname{fix}}K\times C$ [@Bo85 Proposition 11.14] and the lower semicontinuity of $F\cap K$ follows from Proposition \[pr:lsc intersection\]. Notice that the lower semicontinuity condition ii) assumed in Theorem \[th:existenceQEP\] has been replaced in the last two results by the requirement that the lower sections are open. This is due to two different reasons. In the proof of Theorem \[th:sufficientconditions1\], in order to apply [@Yu98 Theorem 1.12] and get that ${\operatorname{gph}}\hat F$ is open, it would be enough to require the lower semicontinuity of $\hat F$. However such an assumption would not guarantee the lower semicontinuity of $F=\hat F\cap C$ which is assumption ii) in Theorem \[th:existenceQEP\]. On the other hand, assumption ii) in Theorem \[th:sufficientconditions2\] is necessary to get the openness of ${\operatorname{gph}}F$ as a consequence of [@Bo85 Proposition 11.14]. The next example shows that a set-valued map $\Phi$ acting from a topological vector space to a polytope $C$ may not have open graph and [@Bo85 Proposition 11.14] fails even if it is lower semicontinuous with convex and open values. Let $C:=\{(x,y)\in {{\mathbb R}}^2:|x|+|y|\leq 1\}$ be a closed convex set in ${{\mathbb R}}^2$. The set-valued map $\Phi:[0,1]\rightarrow C$ defined by $$\Phi(t):=\left\{\begin{array}{ll} C\setminus \{(x,y):x+y=1\} & \mbox{ if } t>0\\ C & \mbox{ if } t=0 \end{array}\right.$$ is lower semicontinuous with convex open values in $C$ but it has not open lower sections since $\phi^{-1}(0,1)=\{0\}$. Nevertheless ${\operatorname{gph}}\Phi$ is not open in $[0,1]\times C$ since the sequence $ \{(n^{-1},1-n^{-1},n^{-1})\}\in [0,1]\times C$ does not belong to ${\operatorname{gph}}\Phi$ but its limit $(0,1,0)\in{\operatorname{gph}}\Phi$. We answer in the negative the question posed in [@BePaRa76] where the authors affirm that they do not know whether [@Bo85 Proposition 11.14] can be generalized to the case where $C$ is an arbitrary convex subset of ${{\mathbb R}}^n$. This also explains why we need to extend the domain of $f(x,\cdot)$ from $C$ to an open subset of ${\operatorname{aff}}C$ in Theorem \[th:sufficientconditions1\]. \[ex:graphnotopen\] Let $C\subseteq {{\mathbb R}}^2$ be the closed unit ball. The set-valued map $\Phi:[0,1]\rightrightarrows C$ defined by $$\Phi(x):=\left\{\begin{array}{ll} C\setminus \{(\cos x,\sin x)\} & \mbox{ if } x>0\\ C & \mbox{ if } x=0 \end{array}\right.$$ has open lower sections and convex open values in $C$. Nevertheless ${\operatorname{gph}}\Phi$ is not open in $[0,1]\times C$. Indeed $(1,0)\in\Phi(0)$ and there is no neighborhood $U$ of $(1,0)$ such that $U\cap C\subseteq\Phi(x)$ for $x$ small enough. A second possible approach for the lower semicontinuity of $F\cap K$ could be to show the nonemptiness of the intersection between the interior of $F$ and $K$. Indeed [@BoGeMyOb84 Corollary 1.3.10] affirms that the set-valued map $\Phi_1\cap\Phi_2$ is lower semicontinuous on the topological space $X$ provided that $\Phi_1,\Phi_2:X\rightrightarrows C$ are convex-valued, lower semicontinuous set-valued maps and $$\label{eq:inte} \Phi_1(x)\cap\Phi_2(x)\neq \emptyset\quad\Rightarrow\quad\Phi_1(x)\cap{\operatorname{int}}\Phi_2(x)\neq \emptyset.$$ The following example shows that such result could not be guaranteed (as erroneously stated in [@Yu98 Theorem 1.13]) if the interior is replaced by the relative interior in condition (\[eq:inte\]). Given a set $C\subseteq{{\mathbb R}}^n$, we denote by ${\operatorname{ri}}C$ the relative interior of $C$, namely, ${\operatorname{ri}}C={\operatorname{int}}_{{\operatorname{aff}}C}C$. Let $C\subseteq {{\mathbb R}}^2$ be the closed unit ball and $\Phi_1:[0,1]\rightrightarrows C$ be defined as in Example \[ex:graphnotopen\]. Consider $\Phi_2:[0,1]\rightrightarrows C$ defined by $$\Phi_2(x):=\{(\cos x,\sin x)\}\qquad \forall x\in[0,1].$$ Then $\Phi_2$ is a continuous single-valued map and $\Phi_1$ is convex-valued with open lower sections. Furthermore $$\Phi_1(x)\cap\Phi_2(x)=\left\{\begin{array}{ll} \emptyset & \mbox{ if } x>0\\ \{(1,0)\} & \mbox{ if } x=0 \end{array}\right.$$ and $\Phi_1(0)\cap {\operatorname{ri}}\Phi_2(0)=C\cap \{(1,0)\}=\{(1,0)\}$. Nevertheless $\Phi_1\cap\Phi_2$ is not lower semicontinuous at $0$. Notice that $\Phi_1(x)$ is even open on $C$, for all $x\in[0,1]$. The following is a correct version of [@Yu98 Theorem 1.13]. \[pr:lsc intersection1\] Let $X$ be a topological space, $C\subseteq{{\mathbb R}}^n$ and $\Phi_1,\Phi_2:X\rightrightarrows C$ be lower semicontinuous and convex-valued. Moreover, for all $x\in X$ assume that ${\operatorname{aff}}\Phi_2(x)={\operatorname{aff}}C$ and $$\Phi_1(x)\cap\Phi_2(x)\neq \emptyset\quad \Rightarrow\quad\Phi_1(x)\cap{\operatorname{ri}}\Phi_2(x)\neq \emptyset$$ then $\Phi_1\cap \Phi_2$ is lower semicontinuous. [**Proof.**]{} By definition, up to isomorphism, there exists $m\leq n$ such that ${\operatorname{aff}}C=x_0+{{\mathbb R}}^m$, where $x_0\in C$ is arbitrarily fixed. Define $\hat\Phi_i:X\rightrightarrows {{\mathbb R}}^m$ by $\hat\Phi_i:=\Phi_i-x_0$, $i=1,2$. Then $\hat\Phi_1$ and $\hat\Phi_2$ are lower semicontinuous and convex-valued. Furthermore, since ${\operatorname{aff}}\Phi_2(x)={\operatorname{aff}}C$, then ${\operatorname{ri}}\Phi_2(x)=x_0+{\operatorname{int}}\hat\Phi_2(x)$ and $\hat\Phi_1(x)\cap {\operatorname{int}}\hat\Phi_2(x)\neq \emptyset$ whenever $\hat\Phi_1(x)\cap \hat\Phi_2(x)\neq \emptyset$. By [@BoGeMyOb84 Corollary 1.3.10] it follows that $\hat\Phi_1\cap \hat\Phi_2$ is lower semicontinuous. This means in turn that $\Phi_1\cap \Phi_2$ is lower semicontinuous. Now we are in position to prove our last existence result. \[th:sufficientconditions3\] Assume that $K$ is lower semicontinuous with nonempty convex values and ${\operatorname{fix}}K$ is closed. Moreover suppose that 1. $F$ is convex-valued on ${\operatorname{fix}}K$, 2. $F$ is lower semicontinuous on ${\operatorname{fix}}K$, 3. ${\operatorname{aff}}K(x)={\operatorname{aff}}C$, for all $x\in \partial_C {\operatorname{fix}}K$, 4. $F(x)$ is open on $C$, for all $x\in \partial_C {\operatorname{fix}}K$, where $F$ is the set-valued map given in (\[eq:mapF\]). Then the quasiequilibrium problem (\[eq:qep\]) has a solution. [**Proof.**]{} It is enough to show that assumption iii) of Theorem \[th:existenceQEP\] holds, i.e. $F\cap K$ is lower semicontinuous on $\partial_C{\operatorname{fix}}K$. Let $x\in\partial_C{\operatorname{fix}}K$ be fixed and assume that $F(x)\cap K(x)\neq\emptyset$ (otherwise the intersection is trivially lower semicontinuous at $x$). By assumption there exists an open set $\Omega\subseteq {{\mathbb R}}^n$ such that $F(x)=\Omega\cap C$. Then $$\emptyset\neq F(x)\cap K(x)=\Omega\cap C\cap K(x)=\Omega\cap K(x).$$ From [@Ro70 Corollary 6.3.2] we get $$\emptyset\neq \Omega\cap {\operatorname{ri}}K(x)=F(x)\cap {\operatorname{ri}}K(x)$$ The lower semicontinuity of $F\cap K$ at $x$ follows from Proposition \[pr:lsc intersection1\]. Now we make a comparison with an analogous result in [@Cu95]. The assumptions of Theorem \[th:sufficientconditions3\] are the same as those of [@Cu95 Theorem 3.2] except that conditions iii) and iv) must be verified for all $x\in \partial_C{\operatorname{fix}}K$ instead of for all $x\in C$. Thus, Theorem \[th:sufficientconditions3\] is clearly more general and, unlike [@Cu95 Theorem 3.2], it reduces to Ky Fan minimax inequality when the constraint set-valued map $K$ is equal to $C$. Conclusions =========== In this paper existence results for the solution of finite dimensional quasiequilibrium problems are obtained by using a Michael selection result for lower semicontinuous set-valued maps. The peculiarity of our results, which make them different from other results in the literature to the best of knowledge of the authors, is the fact that they reduce to Ky Fan minimax inequality when the constraint map is constant. Moreover we provide information regarding the position of a solution. In fact either it is a fixed point of the constraint set-valued map which solves an equilibrium problem or it lies in the boundary of the fixed points set. To know this property seems promising for the construction of solution methods. Future works could be devoted to exploit such result to propose computational techniques for solving quasiequilibrium problems. Another possible advance consists in studying conditions which permit to replace the compactness of the domain with suitable coercivity conditions on the equilibrium bifunction. [00]{} Fan K.: A minimax inequality and applications. In: Shisha O. (ed.): Inequalities III, pp. 103–113. 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--- abstract: 'We observed atmospheric gamma-rays around 10 GeV at balloon altitudes (15$\sim$25 km) and at a mountain (2770 m a.s.l). The observed results were compared with Monte Carlo calculations to find that an interaction model (Lund Fritiof1.6) used in an old neutrino flux calculation was not good enough for describing the observed values. In stead, we found that two other nuclear interaction models, Lund Fritiof7.02 and dpmjet3.03, gave much better agreement with the observations. Our data will serve for examining nuclear interaction models and for deriving a reliable absolute atmospheric neutrino flux in the GeV region.' author: - 'K. Kasahara' - 'E. Mochizuki' - 'S. Torii' - 'T. Tamura' - 'N. Tateyama' - 'K. Yoshida' - 'T. Yamagami' - 'Y. Saito' - 'J. Nishimura' - 'H. Murakami' - 'T. Kobayashi' - 'Y. Komori' - 'M.Honda' - 'T. Ohuchi' - 'S. Midorikawa' - 'T. Yuda' bibliography: - 'betsgamma.bib' title: 'Atmospheric gamma-ray observation with the BETS detector for calibrating atmospheric neutrino flux calculations' --- Introduction ============ The discovery of evidence for neutrino oscillation by the Super Kamiokande group[@skoscillation] is based on the comparison of the observed atmospheric neutrino flux with calculated values. Although the conclusion is so derived that it would not be upset by the uncertainty of the absolute flux value, it is desirable to obtain a reliable expected neutrino flux (under no oscillation assumption) for further detailed discussions. Two major sources of uncertainty in the atmospheric neutrino flux calculation are 1) the primary cosmic-ray spectrum and 2) the propagation of cosmic rays in the atmosphere, especially, modeling of the nuclear interaction. The absolute flux calculations so far made by various groups are expected to have uncertainty of $\sim$ 30 %[@GHreview]. The primary proton and He spectra recently measured with magnet spectrometers by the BESS [@bess1ry] and AMS[@ams1ry] groups agree very well and seem reliable. Therefore, we may take that the first problem mentioned above have now been almost settled at least up to 100 GeV/n. This means that if we have a reliable atmospheric cosmic-ray flux data, we may compare it with a calculation which uses such primaries and test the validity of nuclear interaction models. For such an atmospheric cosmic-ray component, one may first raise the muon and actually some new observations have been or being tried[@capricemuon1; @capricemuon2; @bessmuonnori]. As a secondary cosmic-ray component, we focused on gamma-rays which are easy to measure with our detector. A good model should be able to explain muons and gamma-rays simultaneously. Muons are important since they are directly coupled with neutrinos, but the flux is affected somehow by the structure of the atmosphere which is usually not well known. Compared to muons, the flux of gamma-rays is substantially lower but is almost insensitive to the atmospheric structure and depends only on the total thickness to the observation height. In 1998, we performed first gamma-ray observation with our detector at Mt. Norikura (2770m a.s.l) in Japan, and also made subsequent two successful observations at balloon altitudes (15 $\sim 25$ km) in 1999 and 2000. In the present paper, we report the final results of these observations and consequences. The Detector ============ For our observation, we upgraded the BETS (Balloon-born Electron Telescope with Scintillating fibers) detector which had been developed for the observation of cosmic primary electrons in the 10 GeV region. Its details before being upgraded for gamma-ray observation is in [@betsnim] and the electron observation result is in [@betselec]. The basic performance was tested at CERN using electron, proton and pion beams of 10 to 200 GeV[@betsnim; @betscern]. Although this was undertaken before the upgrading, we can essentially use that calibration for the current observeions partly with a help of Monte Carlo simulations. Figure \[det\] shows a schematic structure of the main body of BETS. The calorimeter has 7.1 r.l lead thickness and the cross-section is 28 cm $\times$ 28 cm. The whole detector system is contained in a pressure vessel made of thin aluminum. ![Schematic illustration of the main body of the detector. S1, S2 and S3 are 1 cm thick plastic scintillators used for trigger. Each fiber has 1mm diameter. Originally nuclear emulsion plates were placed on the upper scifi’s and also inserted between the upper thin lead plates for detailed investigation of tracking capability of scifi. They are kept in the present system to have the same structure at the calibration time. The inlaid cascade shows charged particle tracks by a simulation for a 30 GeV incident proton. \[det\]](detconfigwithshower.eps){width="92mm"} R.M.S energy resolution(%) 21, 18, 15 (for $\theta\sim 15^\circ$) ---------------------------------- ------------------------------------------- S$\Omega$(cm$^2$sr) 243, 240,218 (at $\sim$20 km) R.M.S angular resolution (deg) 2.3, 1.3, 1.0 (for $\theta\sim 15^\circ$) Total number of scifi’s 10080 Weght including electronics (kg) 230 Cross-section of the main body 28cm $\times $ 28cm Thickness (Pb radiation length) 7.1 : Basic characteristics of BETS\ (triple numbers in the table are for gamma-ray energy of 5, 10, and 30 GeV, respectively) \[basicchara\] The main feature of the BETS detector is that it is a tracking calorimeter; it contains a number of sheets consisting of 1 mm diameter scintillating fibers (scifi), many of which are sandwiched between lead plates. The total number of scifi’s are 10080. The sheets are grouped into two types; one is to serve for x and the other for y position measurement. Each of them is fed to an image intensifier which in turn is connected to a CCD. Thus, the two CCD output gives us an $x-y$ image of cascade shower development and enables us to discriminate gamma-rays, electrons from other (mainly hadronic) background showers. The proton rejection power against electron is $R\sim 2\times 10^3$ (i.e, one misidentification among $R$ protons) at 10 GeV[^1] The basic characteristics of the detector are summarized in Table \[basicchara\]. ![Image of cascade shower by a proton (120 GeV,left) and an electron(10 GeV, right) obtained at CERN. \[image\]](showerimage.eps){width="85mm"} In Fig.\[image\], we show examples of the CCD image of a cascade shower for a proton incident case and for an electron incident case. Figure \[anti\] illustrates the yearly change of anti-counters. In 1998 (Mt.Norikura observation), the main change was limited to the upgrading of trigger logic. In 1999, we added 4 side anti-counters (each 15 cm $\times$ 36 cm $\times$ 1.5 cm plastic scintillator. Nine optical fibers containing wave length shifter are embedded in each scintillator and connected to a Hamamatu H6780 PMT. ![Yearly change of the anti-counters. Left: 1998. No change from original BETS except for trigger logic. Middle: 1999. 1.5 cm thick plastic scintillator side anti-counters were added. Right: 2000. The whole top view was covered by a 1 cm thick plastic sintillator. \[anti\]](yearlychange.eps){width="85mm"} In 2000, we further added an anti-counter which covers the whole top view of the detector and also improved data acquisition speed. The top anti-counter is 38 cm $\times$ 38 cm $\times$ 1 cm plastic scintillator. We also embedded optical fibers; 8 in the $x$ and another 8 in the $y$ direction, all of which were fed to an H6780. Although we could remove background showers without the anti-counters, inclined particles (mainly protons) entering from the gap between top scintillator (S1) and the main body degrades the desired gamma-ray event rate. The addition of the top anti-counter greatly helped improve this rate. We emphasize that detection of gamma-rays is easier for us than that of electrons, since, for gamma-rays, we can utilize absence of incident charge. Observations ============ Table \[sumtab\] shows the summary of the observations. Observation Mt.Norikura(1998) --------------------- -------------------- ------ ------- ------- ------- ------- ------- ------- ------- ------- Period Aug.31$\sim$Sep.18 Altitude(km) 2.77 15.3 18.5 21.2 24.7 32.3 15.3 18.3 21.4 25.1 Depth(g/cm$^2$) 737 126 74.8 48.9 28.0 9.5 128 73 45.7 25.3 Obs. hour (s) $1.33\times 10^6$ 1260 1560 2100 4878 3120 1560 2160 4320 2320 Live time (s) $9.8\times 10^5$ 504 450 414 852 498 752 928 1805 789 Live time (%) 74.0 40.0 28.8 19.7 17.5 16.0 48.2 43.0 42.6 44.2 Triggered events $1.8\times 10^6 $ 9513 11288 13361 30439 16741 18808 25795 46675 17436 $\gamma$ events $4.7\times 10^4$ 700 650 611 848 345 1300 1485 2299 740 (%) 2.5 7.3 5.7 4.6 2.8 2.0 6.9 5.8 4.9 4.2 g-low trigger S1 $< 0.5$ condition (in mip). S2 $> 2.3$ S3 $> 1.7$ - Mt. Norikura observation. Our first gamma-ray observation was performed in 1998 at Mt.Norikura Observatory of Univ. of Tokyo, Japan (2770 m a.s.l, latitude 36.1$^\circ$N, longitude 137.55$^\circ$E, magnetic cutoff rigidity $\sim$ 11.5 GV). The atmospheric pressure during the observation is shown in Fig.\[noripress\]. The average atmospheric depth is 737 g/cm$^2$. ![Pressure change during Mt. Norikura observation. The last pressure drop is due to a typhoon. The average pressure is 723 hP (737 g/cm$^2$). \[noripress\]](norikurapressure.eps){width="85mm"} - Balloon flight We had two similar balloon filights in 1999 and 2000. Since the main outcome of the data is from the latter, we briefly describe it. A balloon of 43$\times 10^3$ m$^3$ was launched at 6:30 am, 5th June, 2000 from the Sanriku balloon center of the Institute of Space and Astronautical Science, Japan (latitude 39.2$^\circ$N, longitude 141.8$^\circ$E, magnetic cutoff rigidity $\sim$ 8.9 GV) and recovered with the help of the helicopter. at 17:59 on the sea not far from the center. The flight curve shown in Fig.\[flight\] confirms that we have good level flights at 4 different heights. As compared to the 1999 flight, this flight realized a smaller dead time and higher ratio of desired gamma-ray events. ![Flight curve of the 2000 observation. Pressure (upper) and altitude (lower) as a function of time. Each arrow shows the level flight region. The pressure change at around 15.3 km is rather rapid but the gamma-ray intensity is almost constant there and the change can be neglected. \[flight\]](flightcurve.eps){width="73mm"} Event trigger ------------- The basic event trigger condition is created by signals from the three plastic scintillators (S1, S2 and S3). We show the discrimination level in terms of the minimum ionizing particle number which is defined by the peak of the energy loss distribution of cosmic-ray muons passing both S1 and S3 with inclination less than 30 degrees. We prepare a multi-trigger system by which event trigger with different conditions is possible at the same time. The major two trigger modes are the g-low and g-high. The g-low is responsible for low energy gamma-rays and all anti-counters, when available, are used as veto counters. Its condition is listed in Table \[sumtab\]. High energy gamma-rays normally produce a lot of back splash particles which hit S1 and/or anti-counters, and thus the g-low trigger is suppressed. In such a case, i.e, if we have a large S3 signal, anti-counter veto is invalidated and the S1 threshold is relaxed (The g-high condition is S1$<3.0$, S2$>5.0$ and S3$>8.1$). The branch even point of the g-low and g-high mode efficiency is at $\sim $30 GeV. Since we deal with gamma-rays mostly below 30 GeV, and also to avoid complexity, we present results only by the g-low mode. Analysis ======== Event selection --------------- Among the triggered events, we selected gamma-ray candidates by imposing the following conditions: ![(left)Energy concentration distribution at 21.4 km. (right)the same by electrons at CERN []{data-label="conc"}](Econc.eps){width="8.5cm"} 1. The estimated shower axis passes S1 and S3. The axis position in S3 must be at least 2 cm apart from the edge of S3. 2. The estimated shower axis has a zenith angle less than 30 degrees. 3. The energy concentration (see below) must be greater than 0.7. According to a simulation, only neutrons could be a background against gamma-rays and the 3rd conditions above reduces the neutron contribution to a negligible level ($<1$%). The energy concentration is defined as the fraction of scintillating fiber light intensity within 5 mm from the shower axis. Figure \[conc\] shows the concentration of analysed events together with the result of CERN data. Hadrons make a distribution with a peak at around 0.5. We see that the contribution of hadrons in our observation is negligible. Energy Determination -------------------- The energy calibration was performed in 1996 at CERN using electrons with energy 10 $\sim $ 200 GeV[@betsnim; @betscern]. There is no direct calibration for gamma-rays, but, for the present detector thickness and energy range, a M.C simulation tells us that the calibration in 1996 can be used for gamma-rays, too[^2]. Therefore, for the 1998 and 1999 observations, energy is obtained as a function of the S3 output and zenith angle using the CERN calibration. In 2000, we made some change in the electronics so the CERN calibration could not be used directly. The effect by the change was absorbed by a M.C simulation of which the validity was verified by examining the 1998 and 1999 data. We used the sum of S2 and S3 outputs below 20 GeV since the energy resolution was found to be better than using S3 only. Figure \[eresol\] shows r.m.s energy resolution. ![R.m.s energy resolution. The resolution by S2+S3 or S3 only is shown. Different symbols indicate different incident angles. We used S2+S3 below 20 GeV for the year 2000 data. \[eresol\]](Eres.eps){width="8cm"} Correction of the gamma-ray intensity ------------------------------------- The gamma-ray vertical flux is obtained from the raw $dN/dE$ by dividing it by the live time of the detector and the effective $S\Omega$ (area $\times$ solid angle). The latter is obtained by a simulation[@someganu00]. It is dependent on the observation hight and energy. A typical value at 10 GeV is 240 cm$^2$sr (see Table\[basicchara\]). The energy spectrum is further corrected by the following factors which are not taken into account in the $S\Omega$ calculation. ![(upper)Multiple incidence rate. (lower) Correction factor for year 2000 due to spillover. The flux must be lowered. For Norikura, the factor below 20 GeV is larger by 1$\sim 3$ %. []{data-label="correc"}](turehuta.eps "fig:"){width="7cm"} ![(upper)Multiple incidence rate. (lower) Correction factor for year 2000 due to spillover. The flux must be lowered. For Norikura, the factor below 20 GeV is larger by 1$\sim 3$ %. []{data-label="correc"}](ER_hosei.eps "fig:"){width="7cm"} 1. Systematic bias in our estimation of the shower axis. We underestimate the zenith angle systematically and it leads to overestimation of the intensity about 4% for the balloon and 1.8 % for Mt.Norikura observations. 2. Multiple incidence of particles. A gamma-ray is sometimes accompanied by other charged particles and they enter the detector simultaneously (within 1 ns time difference in 99.9 % cases). They are a family of particles generated by one and the same primary particle[^3]. The charged particles fire the anti-counter and the g-low trigger is inhibited. In some case, multiple gamma-rays enter the detector simultaneously. The rate is smaller than the charged particle case. However, this is judged as a hadronic shower in most of cases. The multiple incidence leads to the underestimation of gamma-ray intensity. The portion of multiple incidence is shown in Fig.\[correc\] (upper). 3. Finite energy resolution. The rapidly falling energy spectrum leads to the spillover effect. This normally leads to the overestimation of flux (Fig.\[correc\], lower). Results and comparison with calculations ======================================== The flux values are summarized in Table \[flux\]. We put only the statistical errors in the flux values, since systematic errors coming from the uncertainty of the S$\Omega$ calculation, various cuts and flux corrections are expected to be order of a few percent and much smaller than the present statistical errors. [|l|l|l|l|l|l|l|l|l|l|]{}\ & & & &\ \ 5.48 & 2.42 $\pm$ 0.37 & 5.48 & 2.11 $\pm$ 0.39 & 5.47 & 2.11 $\pm$ 0.24 & 5.47 & 1.58 $\pm$ 0.25 & 5.47 & 0.49 $\pm$ 0.14\ 6.47 & 1.18 $\pm$ 0.27 & 6.47 & 1.10 $\pm$ 0.24 & 6.47 & 1.35 $\pm$ 0.21 & 6.47 & 0.82 $\pm$ 0.18 & 6.57 & 0.19 $\pm$ 0.09\ 7.47 & 0.89 $\pm$ 0.24 & 7.47 & 0.79 $\pm$ 0.21 & 7.47 & 0.82 $\pm$ 0.16 & 7.47 & 0.66 $\pm$ 0.16 & 7.47 & 0.24 $\pm$ 0.10\ 8.48 & 0.37 $\pm$ 0.15 & 8.48 & 0.92 $\pm$ 0.20 & 8.48 & 0.51 $\pm$ 0.13 & 8.48 & 0.49 $\pm$ 0.14 & 8.48 & 0.16 $\pm$ 0.08\ 9.48 & 0.54 $\pm$ 0.17 & 9.85 & 0.46 $\pm$ 0.11 & 9.48 & 0.50 $\pm$ 0.12 & 9.48 & 0.36 $\pm$ 0.12 & 9.48 & 0.16 $\pm$ 0.08\ 10.5 & 0.17 $\pm$ 0.10 & 11.5 & 0.35 $\pm$ 0.12 & 10.5 & 0.41 $\pm$ 0.09 & 10.5 & 0.34 $\pm$ 0.12 & 12.3 & 0.13 $\pm$ 0.037\ 12.1 & 0.28 $\pm$ 0.09 & 14.0 & 0.24 $\pm$ 0.06 & 11.8 & 0.23 $\pm$ 0.069 & 12.2 & 0.21 $\pm$ 0.054 & 17.0 & 0.032 $\pm$ 0.018\ 14.0 & 0.17 $\pm$ 0.05 & 18.3 & 0.072 $\pm$ 0.030 & 14.0 & 0.16 $\pm$ 0.030 & 14.0 & 0.076 $\pm$ 0.03 & 21.7 & 0.022$\pm$ 0.015\ 18.5 & 0.12 $\pm$ 0.04 & 26.8 & 0.040 $\pm$ 0.017 & 18.4 & 0.086 $\pm$ 0.023& 17.8 & 0.078 $\pm$ 0.029 & &\ 25.5 & 0.06 $\pm$ 0.02 & & & 27.1 & 0.026 $\pm$ 0.009& 21.7 & 0.064 $\pm$ 0.026 & &\ & & & & & & 26.8 & 0.024 $\pm$ 0.012 & &\ & & & & & & 36.0 & 0.012 $\pm$ 0.008 & &\ E(GeV) Flux ($10^{-4}/$m$^2\cdot$s$\cdot$sr$\cdot$GeV) -------- ------------------------------------------------- 5.48 274 $\pm$ 13 6.47 183 $\pm$ 11 7.47 133 $\pm$ 9 8.47 87.8 $\pm$ 7.5 9.47 86.5 $\pm$ 7.5 10.5 54.1 $\pm$ 5.9 11.5 46.6 $\pm$ 5.5 12.5 38.3 $\pm$ 5.0 13.5 32.6 $\pm$ 4.6 14.5 24.2 $\pm$ 4.0 15.5 25.7 $\pm$ 4.1 17.0 11.9 $\pm$ 2.0 19.0 15.3 $\pm$ 2.3 21.0 13.1 $\pm$ 2.1 23.0 5.80 $\pm$ 1.4 26.0 5.31 $\pm$ 0.95 30.0 3.00 $\pm$ 0.72 34.0 2.30 $\pm$ 0.64 38.0 1.07 $\pm$ 0.44 45.0 1.45 $\pm$ 0.32 55.0 0.52 $\pm$ 0.20 65.0 0.22 $\pm$ 0.13 75.0 0.30 $\pm$ 0.15 85.0 0.15 $\pm$ 0.10 : Flux values at Mt. Norikura\[noriflux\] The gamma-ray energy spectra thus obtained at balloon altitudes are shown in Fig.\[balspec\] together with the expected ones calculated by the Cosmos simulation code[@cosmos]. Except for 32.3 km altitude, we can disregard the small difference of the observation depths and we combine two flight data with statistical weight, although the main contribution is from the flight in 2000. In the simulation calculation, we employed 3 different nuclear interaction models: 1) fritiof1.6[@oldfri] used in the HKKM calculation[@hkkm95], which was widely used for comparison with the Kamioka data, 2)fritiof7.02[@newfri][^4] and 3) dpmjet3.03[@dpmjet]. As the primary cosmic ray, we used the BESS result on protons and He. The CNO component is also considered[@cno]. Besides these we included electron and positron data by AMS[@amselec]. Their data in the 10 GeV region is consistent with the HEAT[@heat] and BETS[@betselec] data. Bremstrahlung gamma-rays from the primary electrons could contribute order of $\sim 10$ % at very high altitudes. At balloon altitudes, the two models, fritiof7.02 and dpmjet3.03, give almost the same results which are close to the observed data, while fritof1.6 gives clearly smaller fluxes than the observation. Figure \[norispec\] shows the result from the observation at Mt.Norikura. It should be noted that the flux by fritiof1.6 becomes higher than the ones by the other models at this altitude. From these figures, we see fritiof7.02 and dpmjet3.03 give rapider increase and faster attenuation of intensity than fritiof1.6; the tendency is very consistent with the observed data. The transition curve of the flux integrated over 6 GeV shown in Fig.\[transition\] clearly demonstrates this feature. {width="6.5cm"} {width="6.5cm"} {width="6.5cm"} {width="6.5cm"} ![Gamma-ray spectra at 5 balloon heights are compared with 3 different models. The vertical axis is Flux$\times E^2$. Except for 1999 data at 32.3 km, 1999 and 2000 flights data are combined. From top to bottom, at 25.1, 21.4, 18.3, 15.3 and 32.3 km. The spectra expected from three interaction models are drawn by solid (dpmjet3.03), dash (fritiof7.02) and dotted (fritiof1.6) lines. []{data-label="balspec"}](spectrum5.eps){width="6.5cm"} ![Gamma-ray spectrum at Mt. Norikura (2.77 km a.s.l). The vertical axis is Flux$\times E^2$. Our data is at $<$ 100 GeV. Data above 300 GeV is from emulsion chamber experiments. For the latter, see Sec.\[discuss\] []{data-label="norispec"}](norikura.eps){width="7.5cm"} ![The altitude variation of the flux integrated over 6 GeV. The dpmjet3.03 and fritiof7.02 give almost the same feature consistent with the observation while the deviation of fritiof1.6 from the data is obvious. \[transition\]](transition.eps){width="7.5cm"} Discussions\[discuss\] ====================== Comparison with other data -------------------------- We found Fritiof7.02 and dpmjet3.03 give good agreement with the observed gamma-ray data at around 10 GeV. We briefly see whether these models can interpret other observations. More detailed inspection will be done elsewhere. - Muon data by the BESS group at Mt.Norikura[@bessmuonnori]. Recently, the BESS group reported detailed muon spectrum over several hundred MeV/c. In their paper, calculations by dpmjet3.03 and fritiof1.6 are compared with the data; agreement by dpmjet3.03 is quit good at least above GeV where Fritiof7.02 also gives more or less the same flux. On the other hand, fritiof1.6 shows too high flux. These features are consisten with our present analysis. - Higher energy gamma-ray data by emulsion chamber. In Fig. \[norispec\], we inlaid an emulsion chamber data[@ecc][^5] at Mt. Norikura. Our data seems to be smoothly connected to their data as the two interaction models (Fritiof7.02 and dpmjet3.03) predict. Since the emulsion chamber data extends to the TeV region and the primary particle energy responsible for such high energy gamma-rays is much higher than 100 GeV where we have no accurate information comparable to the AMS and BESS data, it would be premature to draw a definite conclusion on the primary and interaction model separately. However, the fact that smooth extrapolation of the primary spectra as shown in Table \[extendprim\] and the interaction model, dpmjet3.03 or fritiof7.02, give a consistent result with the data, seems to indicate that such combination would provide a good estimate on other components at $\gg$ 10 GeV. ------- ----------- ------- ----------- -------- ---------   E    flux    E    flux    E   flux 92.6 0.593E-01 79.4 0.549E-02 100. 9.0E-5 108 0.388E-01 100. 3.0E-3 400. 1.8E-6 126 0.276E-01 200. 5.0E-4 2.0E3 3.5E-8 147 0.179E-01 400. 7.0E-5 2.0E4 9.3E-11 171 0.124E-01 2.0E3 9.98E-7 2.0E5 2.3E-13 200 0.836E-02 2.0E4 2.5E-9 14.0E5 1.3E-15 1100 8.29E-5 2.0E5 3.97E-12 3.0E6 1.7E-16 1.1E4 1.47E-7 4.0E5 6.1E-13 3.0E7 2.0E-19 1.1E5 2.8E-10 8.0E5 7.0E-14 3.0E8 2.2E-22 2.2E5 3.7E-11 8.0E6 8.7E-17 4.4E5 5.0E-12 8.0E8 5.3E-23 4.4E8 2.8E-21 ------- ----------- ------- ----------- -------- --------- : Primary flux assumed in the simulation above 100 GeV/n\ (E in kinetic energy per nucleon (GeV), flux in /m$^2\cdot$s$\cdot$sr$\cdot$GeV) \[extendprim\] The $x$-distributions --------------------- The two models, fritiof7.02 and dpmjet3.03, give almost the same results in the present comparison. However, if we look into the $x$-distribution of the particle production, we note some difference, especially in the proton $x$-distribution. We define the $x$ as the kinetic energy ratio of the incoming proton and a secondary particle in the laboratory frame. The $x$ distribution for $p$Air collisions at incident proton energy of 40 GeV is presented for photons (from $\pi^0$ plus $\eta$ decay) and protons in Fig.\[xdist\]. Difference of the three models seen in the photon distribution is quite similar to the one for charged pions. The $x$ region most effective to atmospheric gamma-ray flux is around 0.2$\sim$0.3 where the difference is not so large but fritiof7.02 and dpmjet3.03 have higher gamma-ray yield than fritiof1.6. ![The $x$-distribution of photons from $\pi^0$ plus $\eta$ decay (upper) and protons (lower) for $p$Air collisions at 40 GeV. The three model results are shown. []{data-label="xdist"}](gammaxdist.eps "fig:"){width="7.5cm"} ![The $x$-distribution of photons from $\pi^0$ plus $\eta$ decay (upper) and protons (lower) for $p$Air collisions at 40 GeV. The three model results are shown. []{data-label="xdist"}](protonxdist.eps "fig:"){width="7.5cm"} On the other hand, the proton $x$ distribution has larger difference among the three models (we note, however, the difference may be exaggerated than the photon case due to the scale difference). It is interesting to see that, in spite of these large differences, the final flux is not so much different each other. Our gamma-ray data prefers to rather more inelastic feature of collisions than fritiof1.6, i.e rapider increase and faster attenuation of the flux. We should compare the distribution with accelerator data; however, there is meager stuff appropriate for our purpose. One such comparison has been done in a recent review paper[@GHreview] for $p$Air collisions at 24 GeV/c incident momentum. The charged pion distribution by fritiof1.6 and dpmjet3.03 well fit to some scattered data which prevents to tell the superiority of the two. As to the proton distribution, among the three models, fritiof1.6 is rather close to the data but deviation from the data is much larger than the pion case. The proton $x$-distribution would strongly affect the atmospheric proton spectrum. We calculated proton flux at Mt.Norikura to find a flux relation such that fritiof1.6 $>$ fritiof7.02 $>$ dpmjet3.03 as expected naturally from the $x$-distributions. The maximum difference is factor $\sim 2.5$ in the energy region of 0.3 to 3 GeV. The BESS group has measured the proton spectrum at Mt. Norikura in the same energy region. Their result expected to come soon[@sanukibess] will help select a better model for the proton $x$ distribution. summary ======= - We have made successful observation of atmospheric gamma-rays at around 10 GeV at Mt.Norikura (2.77 km a.s.l) and at balloon altitudes (15 $\sim$ 25 km). - The observed gamma-ray fluxes are compared with calculations by three interaction models; it is found that fritiof1.6 employed by the HKKM calculation [@hkkm95], which was used in comparison with the Kamioka data, is not a very good model. - Other two models (fritiof7.02 and dpmjet3.03) give better results consistent with the data, which shows rapider increase and faster attenuation of the flux than fritiof1.6 predicts. - Our data has complementary feature to muon data and will serve for checking nuclear interaction models used in atmospheric neutrino calculations. We sincerely thank the team of the Sanriku Balloon Center of the Institute of Astronautical Science for their excellent service and the support of the balloon flight. We also thank the staff of the Norikra Cosmic-Ray observatory, Univ. of Tokyo. for their help. We are also indebted to S.Suzuki, P.Picchi, and L. Periale for their spport at CERN in the beam test. For the management of X5 beam line of SPS at CERN, we would like to thank L. Gatignon and the tecnical staffs. One of the authors (K.K) thanks S. Roesler for his help in implementing dpmjet3.03. This work is partly supported by Grants-in Aid for Scientific Research B (09440110), Grants-in Aid for Scientific Research on Priority Area A (12047224) and Grant-in Aid for Project Research of Shibaura Institute of Technology. [^1]: We note electron showers of 10 GeV are normally simulated by $\sim$ 30 GeV protons when the latter start cascade at a shallow depth of the detector. [^2]: If we don’t impose the trigger condition, the gamma-ray case shows a small difference from the electron case. [^3]: The chance coincidence probability of uncorrelated particles is negligibly small. [^4]: It is used at energies greater than 10 GeV. At lower energies, model is the same as fritiof1.6 [^5]: Electrons included in the original data is subtracted statistically by use of cascade theory which is accurate at high energies.

--- abstract: 'The maximal minors of a $p\times (m + p)$-matrix of univariate polynomials of degree $n$ with indeterminate coefficients are themselves polynomials of degree $np$. The subalgebra generated by their coefficients is the coordinate ring of the quantum Grassmannian, a singular compactification of the space of rational curves of degree $np$ in the Grassmannian of $p$-planes in ($m + p$)-space. These subalgebra generators are shown to form a sagbi basis. The resulting flat deformation from the quantum Grassmannian to a toric variety gives a new “Gröbner basis style” proof of the Ravi-Rosenthal-Wang formulas in quantum Schubert calculus. The coordinate ring of the quantum Grassmannian is an algebra with straightening law, which is normal, Cohen-Macaulay, Gorenstein and Koszul, and the ideal of quantum Plücker relations has a quadratic Gröbner basis. This holds more generally for skew quantum Schubert varieties. These results are well-known for the classical Schubert varieties $(n=0)$. We also show that the row-consecutive $p\times p$-minors of a generic matrix form a sagbi basis and we give a quadratic Gröbner basis for their algebraic relations.' address: - | - Frank Sottile\ Department of Mathematics\ University of Wisconsin\ Madison, Wisconsin 53706\ USA - | - Bernd Sturmfels\ Department of Mathematics\ University of California\ Berkeley, California 94720\ USA author: - Frank Sottile - Bernd Sturmfels title: A sagbi basis for the quantum Grassmannian --- [^1] Statement of the main result ============================ Let ${\mathcal M}(t)$ be the $p\times(m+p)$-matrix whose $i,j$th entry is the degree $n$ polynomial in $t$, $$x_{i,j}^{(n)} \cdot t^n \ + \, x_{i,j}^{(n-1)} \cdot t^{n-1} +\,\, \ldots \,\, + \, x_{i,j}^{(2)} \cdot t^2 \ + \, x_{i,j}^{(1)} \cdot t\ + \, x_{i,j}^{(0)} .$$ The coefficients $x_{i,j}^{(l)}$ are indeterminates. We write $k[X]$ for the polynomial ring over a field $k$ generated by these indeterminates, for $i=1,\ldots,p$, $j=1,\ldots,m+p$, and $l=0,\ldots,n$. Lexicographic order on the triples $l,i,j$ gives a total order of these variables. For example, $$x_{1,2}^{(0)}\ <\ x_{1,2}^{(1)}\ <\ x_{1,5}^{(1)}\ <\ x_{2,3}^{(1)}\ <\ x_{2,4}^{(1)}\ <\ x_{1,3}^{(2)} \ .$$ Let $\prec$ be the resulting degree reverse lexicographic term order on the polynomial ring $k[X]$. For each $\alpha\in\binom{[m+p]}{p}$ and $a\geq 0$, let $\alpha^{(a)}$ be a variable which, when $a\leq np$, formally represents the coefficient of $t^a$ in the maximal minor of ${\mathcal M}(t)$ given by the columns indexed by $\alpha_1,\alpha_2,\ldots,\alpha_p$. These variables $\alpha^{(a)}$ have a natural partial order, denoted ${\mathcal C}_{p,m}$, which is defined as follows: $$\alpha^{(a)}\ \leq\ \beta^{(b)} \quad\Longleftrightarrow\quad a\leq b \, \mbox{ and } \,\, \alpha_i\leq \beta_{b-a+i} \, \mbox{ for } i = 1,2,\ldots,p-b+a.$$ Fix $0\leq q\leq np$ and let ${\mathcal C}^q_{p,m}$ denote the truncation of the infinite poset ${\mathcal C}_{p,m}$ to the finite subset $\,\bigl\{ \, \alpha^{(a)} \,\, | \,\, \alpha\in\binom{[m+p]}{p} \mbox{ and } a\leq q \, \bigr\}$. The posets ${\mathcal C}^q_{p,m}$ are graded distributive lattices. Figure \[fig:qorder\] shows ${\mathcal C}^1_{2,3}$. $$\epsfxsize=1.6in \epsfbox{fig1.eps}$$ Let $\varphi : k[{\mathcal C}^q_{p,m}] \rightarrow k[X]$ denote the $k$-algebra homomorphism which sends the formal variable $\alpha^{(a)}$ to the coefficient of $t^a$ in the $\alpha$th maximal minor of the matrix ${\mathcal M}(t)$. For example, [ $$\begin{array}{l} \varphi \bigl( 456^{(2)} \bigr) \quad = \quad \mbox{\rm coefficient of $t^2$ in}\ \ \det\left[\begin{array}{ccc} x^{(0)}_{1,4}+x^{(1)}_{1,4}t&\ x^{(0)}_{1,5}+x^{(1)}_{1,5}t\ & x^{(0)}_{1,6}+x^{(1)}_{1,6}t\\ \rule{0pt}{17pt} x^{(0)}_{2,4}+x^{(1)}_{2,4}t&\ x^{(0)}_{2,5}+x^{(1)}_{2,5}t\ & x^{(0)}_{2,6}+x^{(1)}_{2,6}t\\ \rule{0pt}{17pt} x^{(0)}_{3,4}+x^{(1)}_{3,4}t&\ x^{(0)}_{3,5}+x^{(1)}_{3,5}t\ & x^{(0)}_{3,6}+x^{(1)}_{3,6}t \end{array}\right]\, \\ = \quad \rule{0pt}{20pt} -\underline{x^{(0)}_{3,6}x^{(1)}_{1,5}x^{(1)}_{2,4}} +x^{(0)}_{3,5}x^{(1)}_{1,6}x^{(1)}_{2,4} +x^{(0)}_{3,6}x^{(1)}_{1,4}x^{(1)}_{2,5} -x^{(0)}_{3,4}x^{(1)}_{1,6}x^{(1)}_{2,5} -x^{(0)}_{3,5}x^{(1)}_{1,4}x^{(1)}_{2,6} +x^{(0)}_{3,4}x^{(1)}_{1,5}x^{(1)}_{2,6}\\\rule{0pt}{17pt}\phantom{=} \quad +x^{(0)}_{2,6}x^{(1)}_{1,5}x^{(1)}_{3,4} -x^{(0)}_{2,5}x^{(1)}_{1,6}x^{(1)}_{3,4} -x^{(0)}_{2,6}x^{(1)}_{1,4}x^{(1)}_{3,5} +x^{(0)}_{2,4}x^{(1)}_{1,6}x^{(1)}_{3,5} +x^{(0)}_{2,5}x^{(1)}_{1,4}x^{(1)}_{3,6} -x^{(0)}_{2,4}x^{(1)}_{1,5}x^{(1)}_{3,6}\\\rule{0pt}{17pt}\phantom{=} \quad -x^{(0)}_{1,6}x^{(1)}_{2,5}x^{(1)}_{3,4} +x^{(0)}_{1,5}x^{(1)}_{2,6}x^{(1)}_{3,4} +x^{(0)}_{1,6}x^{(1)}_{2,4}x^{(1)}_{3,5} -x^{(0)}_{1,4}x^{(1)}_{2,6}x^{(1)}_{3,5} -x^{(0)}_{1,5}x^{(1)}_{2,4}x^{(1)}_{3,6} +x^{(0)}_{1,4}x^{(1)}_{2,5}x^{(1)}_{3,6}\,. \end{array}$$]{} \[issagbi\] The set of polynomials $\, \varphi(\alpha^{(a)}) \,$ as $\alpha^{(a)} $ runs over the poset ${\mathcal C}^q_{p,m} \,$ forms a sagbi basis with respect to the reverse lexicographic term order $\prec$ on $\,k[X]\,$ defined above. Our second theorem states that the subalgebra [*image*]{}$(\varphi)$ of $k[X]$ generated by this sagbi basis is an [*algebra with straightening law*]{} on the poset ${\mathcal C}^q_{p,m}$. Let $\prec$ be the degree reverse lexicographic term order on $k[{\mathcal C}^q_{p,m}]$ induced by any linear extension of the poset ${\mathcal C}^q_{p,m}$. This term order on $k[{\mathcal C}^q_{p,m}]$ and the previous term order on $k[X]$ are fixed throughout this paper. \[thm:gbasis\] The reduced Gröbner basis of the kernel of $\varphi$ consists of quadratic polynomials in $\,k[{\mathcal C}^q_{p,m}]\,$ which are indexed by pairs of incomparable variables $\gamma^{(c)},\delta^{(d)}$ in the poset $\,{\mathcal C}^{np}_{p,m}$, $$S(\gamma^{(c)},\delta^{(d)}) \quad = \quad \gamma^{(c)}\cdot\delta^{(d)}\ -\ (\gamma^{(c)}\vee\delta^{(d)})\cdot (\gamma^{(c)}\wedge\delta^{(d)}) \,\, + \,\,\hbox{lower terms in $\prec$},$$ and all lower terms $\,\lambda\beta^{(b)}\alpha^{(a)}\,$ in $\,S(\gamma^{(c)},\delta^{(d)}) \,$ satisfy $\,\beta^{(b)}<\gamma^{(c)}\wedge\delta^{(d)}$ and $\gamma^{(c)}\vee\delta^{(d)}<\alpha^{(a)}$. The join $\vee$ and meet $\wedge$ appearing in the above formula are the lattice operations in ${\mathcal C}^{np}_{p,m}$. The combinatorial structure of this distributive lattice will become clear in Section 2, when we introduce the toric variety associated with ${\mathcal C}^{np}_{p,m}$. In Section 3 we interpret the subalgebra ${\rm image}(\varphi)$ of $k[X]$ as the coordinate ring of the quantum Grassmannian. Section 4 contains the proofs of Theorems 1 and 2. These results generalize the classical sagbi basis property of maximal minors [@Sturmfels_invariant Theorem 3.2.9] and its geometric interpretation as a toric deformation [@Sturmfels_GBCP Proposition 11.10] from the case of the Grassmannian to the quantum Grassmannian. In Section 5 we discuss corollaries, applications and some open problems. One such application is that the row-consecutive $p\times p$-minors of any matrix of indeterminates form a sagbi basis. We thank Aldo Conca, Ezra Miller, and Brian Taylor for their helpful comments. The toric variety of the distributive lattice ============================================= Theorem 1 asserts that the initial algebra of our subalgebra $\,{\rm image}(\varphi)\,$ is generated by the initial monomials of its generators $\varphi(\alpha^{(a)}) $. Our first step is to identify the initial monomials. Here are two examples. The first one is the underlined monomial right before Theorem \[issagbi\]: $$\mbox{\rm in}_\prec \bigl( \varphi(456^{(2)}) \bigr) \ =\ x_{3,6}^{(0)}x_{1,5}^{(1)}x_{2,4}^{(1)}\qquad\mbox{and}\qquad \mbox{\rm in}_\prec \bigl( \varphi(2457^{(5)}) \bigr) \ =\ x_{2,7}^{(1)}x_{3,5}^{(1)}x_{4,4}^{(1)}x_{1,2}^{(2)}.$$ In general, the initial monomial of $\varphi(\alpha^{(a)}) $ is given by the following lemma: \[lemma:leadmon\] Let $\alpha\in\binom{[m+p]}{p}$ and $\,a=pl+r \,$ with integers $p > r \geq 0$. Then $$\mbox{\rm in}_\prec \bigl( \varphi(\alpha^{(a)}) \bigr) \quad =\quad x_{r+1,\alpha_p}^{(l)}x_{r+2,\alpha_{p-1}}^{(l)}\cdots x_{p,\alpha_{r+1}}^{(l)} x_{1,\alpha_r}^{(l+1)} x_{2,\alpha_{r-1}}^{(l+1)} \cdots x_{r,\alpha_1}^{(l+1)}.$$ [**Proof.** ]{} Let $x_{i_1,j_1}^{(l_1)}x_{i_2,j_2}^{(l_2)}\cdots x_{i_p,j_p}^{(l_p)}$ be a monomial which appears in $\varphi(\alpha^{(a)})$. We claim that $$\label{eq:comparison} x_{i_1,j_1}^{(l_1)}x_{i_2,j_2}^{(l_2)}\cdots x_{i_p,j_p}^{(l_p)} \quad \preceq\quad x_{r+1,\alpha_p}^{(l)}x_{r+2,\alpha_{p-1}}^{(l)}\cdots x_{p,\alpha_{r+1}}^{(l)}x_{1,\alpha_r}^{(l+1)}\cdots x_{r,\alpha_1}^{(l+1)}.$$ We may assume $\,x_{i_1,j_1}^{(l_1)} \prec x_{i_2,j_2}^{(l_2)} \prec \cdots \prec x_{i_p,j_p}^{(l_p)}\,$ and hence $l_1 \leq\cdots\leq l_p$. Since $l_1+\cdots+l_p=a$, either $ l_1 < q $, from which (\[eq:comparison\]) follows, or else $l_1=\cdots=l_{p-r}=l$ and $l_{p+1-r}=\cdots=l_p=l+1$. In the second case, as $\{i_1,\ldots,i_p\}=\{1,\ldots,p\}$ and the monomial is in order, we must have $i_1<\cdots<i_{p-r}$ and $i_{p+1-r}<\cdots<i_p$. If $i_1 \leq r$, then (\[eq:comparison\]) follows, and if $ i_1 = r+1$, then the ordered sequence $\,i_1,i_2,\ldots,i_p\,$ equals $\, r \! + \! 1,r \! + \! 2,\ldots,p,1,\ldots,r$. Among all monomials satisfying this new second case, the largest in the degree reverse lexicographic order $\prec $ has the second lower index appearing in reverse order. This completes the proof. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ We next introduce some combinatorics to help understand the poset ${\mathcal C}_{p,m}$. A [*row*]{} with shift $a$ consists of $p$ consecutive empty unit boxes shifted $a$ units to the left of a given vertical line. A [*skew shape*]{} is an array of such rows whose shifts are weakly increasing read from top to bottom. For example, the unshaded boxes in the figure on the left form a skew shape (with shifts 1,2,2, and 5) while those in the other figure do not: $$\epsfxsize=2.8in \epsfbox{fig2.eps}$$ A [*(skew) tableau*]{} $T$ is a filling of a skew shape with integers that increase across each row. When the entries lie in $[m+p]$, the $i$th row of a tableau is a sequence $\alpha_{(i)}\in\binom{[m+p]}{p}$. If $a_i$ is the shift of the $i$th row and $T$ has $j$ rows, then $T$ corresponds to a monomial $\alpha_{(1)}^{(a_1)}\alpha_{(2)}^{(a_2)}\cdots\alpha_{(j)}^{(a_j)}$. Conversely, any monomial in the variables $\alpha^{(a)}$ corresponds to a tableau. A tableau $T$ is [*standard*]{} if the entries are weakly increasing in each column, read top to bottom. Equivalently, $T$ is standard if we have $\alpha_{(1)}^{(a_1)}\leq\alpha_{(2)}^{(a_2)}\leq\cdots \leq\alpha_{(j)}^{(a_j)}$ in ${\mathcal C}_{p,m}$. For example, the following two tableaux correspond to the monomials $345^{(0)}123^{(1)}245^{(3)}$ and $135^{(0)}123^{(1)}257^{(3)}$. The first tableau is not standard and the second tableau is standard: $$\epsfxsize=2.8in \epsfbox{fig3.eps}$$ The elements of the poset ${\mathcal C}^q_{p,m}$ are represented by one-row tableaux with entries in $[m+p]$ and shift at most $q$. Two elements satisfy $\,\alpha^{(a)} \leq \beta^{(b)} \,$ if and only if the two-rowed tableau $T=\alpha^{(a)}\beta^{(b)}$ is standard. This representation implies that ${\mathcal C}^q_{p,m}$ is a distributive lattice. Indeed, the two lattice operations $\wedge$ and $\vee$ are described as follows. If a two-rowed tableau $\,T=\alpha^{(a)}\beta^{(b)}\,$ is non-standard then interchanging the entries in every column in which a violation ($\alpha_{a-b+i}<\beta_i$) occurs yields a standard tableau. The first row of this new tableau is the [*meet*]{} $\alpha^{(a)}\wedge\beta^{(b)}$ of $\alpha^{(a)}$ and $\beta^{(b)}$ in ${\mathcal C}^q_{p,m}$ and the second row is their [*join*]{} $\alpha^{(a)}\vee\beta^{(b)}$. Let $\psi : k[{\mathcal C}^{q}_{p,m}] \rightarrow k[X]$ denote the $k$-algebra homomorphism which sends the variable $\alpha^{(a)}$ to the monomial $\,\mbox{\rm in}_\prec \bigl(\varphi(\alpha^{(a)}) \bigr)$. Its kernel is a [*toric ideal*]{} (i.e. binomial prime)  in $\, k[{\mathcal C}^{q}_{p,m}]$. \[lem:toric\] The reduced Gröbner basis for the kernel of $\psi$ consists of the binomials $$\underline{\alpha^{(a)}\cdot\beta^{(b)}} \ -\ (\alpha^{(a)}\vee\beta^{(b)})\cdot(\alpha^{(a)}\wedge\beta^{(b)}),$$ where $\alpha^{(a)}, \beta^{(b)}$ runs over all incomparable pairs of ${\mathcal C}^q_{p,m} $. The initial monomial is underlined. [**Proof.**]{} This follows from Hibi’s Theorem [@Hibi] since ${\mathcal C}^q_{p,m}$ is a distributive lattice. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ \[cor:standard\] The set of standard tableaux is a $k$-basis for $\, k[{\mathcal C}^q_{p,m}] / {\rm kernel}(\psi) = {\rm image}(\psi) $. Here is a typical element in the reduced Gröbner basis of ${\rm kernel}(\psi) $ for $p=5,m=4,q=9$: $$\underline{45789^{(1)} \cdot 12356^{(3)}} \, - \, 35689^{(1)} \cdot 12457^{(3)}.$$ Note that the second monomial corresponds to a standard tableau while the first does not. We write $T^q_{p,m}$ for the projective toric variety cut out by the binomials in Lemma \[lem:toric\]. Its coordinate ring is the subalgebra $\,{\rm image}(\psi)$ of $k[X]$. The geometry of toric varieties associated with distributive lattices is discussed in [@Wagner]. The analogue to Corollary \[cor:standard\] always holds, i.e., multichains in the poset correspond to basis monomials in the coordinate ring. \[Degree\] The degree of the toric variety $T^q_{p,m}$ is the number of maximal chains in ${\mathcal C}^q_{p,m}$. The closed intervals of the poset ${\mathcal C}^q_{p,m}$ are also distributive lattices. They are denoted $$[\beta^{(b)}, \alpha^{(a)}] \quad := \quad \bigl\{ \gamma^{(c)} \in {\mathcal C}^q_{p,m} \,: \, \beta^{(b)} \leq \gamma^{(c)} \leq \alpha^{(a)} \bigr\}.$$ Proposition \[lem:toric\] and Corollaries \[cor:standard\] and \[Degree\] hold essentially verbatim for the distributive sublattice $\,[\beta^{(b)}, \alpha^{(a)}]\,$ as well. The projective toric variety associated with $\,[\beta^{(b)}, \alpha^{(a)}]\,$ is gotten from the toric variety of ${\mathcal C}^q_{p,m}$ by setting $\gamma^{(c)} = 0$ for all $ \gamma^{(c)} \not\in [\beta^{(b)}, \alpha^{(a)}]$. The degree of that variety is the number of saturated chains in ${\mathcal C}^q_{p,m}$ which start at $\beta^{(b)}$ and end at $ \alpha^{(a)}$. We close this section with an alternative proof, to be used in Section 4, for the fact that ${\mathcal C}_{p,m}$ is a distributive lattice. We claim that ${\mathcal C}_{p,m}$ is a sublattice of [*Young’s lattice*]{}. Given $\alpha^{(a)} \in {\mathcal C}_{p,m}$, write $a=pl+r$ with integers $p > r \geq 0$, and define a sequence $J(\alpha^{(a)})$ by $$\label{eq:seq_def} J(\alpha^{(a)})_i\quad :=\quad \left\{\begin{array}{ll} l(m+p)+\alpha_{r+i}&\quad\mbox{if }1 \leq i\leq p-r\\ (l+1)(m+p)+\alpha_{i-p+r}&\quad\mbox{if }p-r<i\leq p \end{array}\right.\,.$$ This gives an order-preserving bijection between the poset ${\mathcal C}_{p,m}$ and the poset of sequences $J:=j_1<j_2<\cdots<j_p$ of positive integers with $j_p-(m+p)<j_1$, and it preserves meet and join. This bijection preserves the rank function in the two distributive lattices: $$\label{Miracle} | \alpha^{(a)} | \quad := \quad a(m+p) + \sum_{j=1}^p (\alpha_j - j) \quad = \quad \sum_{i=1}^p \bigl( J(\alpha^{(a)})_i - i \,\bigr) \quad =: \quad | J(\alpha^{(a)})|.$$ The quantum Grassmannian ======================== Let ${\it Grass}_pk^{m+p}$ denote the Grassmannian of $p$-planes in the vector space $k^{m+p}$. This is a smooth projective variety of dimension $mp$. Consider the space $S^q_{p,m}$ of maps ${\mathbb P}^1\rightarrow \mbox{\it Grass}_pk^{m+p}$ of degree $q$. Such a map may be (non-uniquely) represented as the row space of a $p\times(m+p)$-matrix of polynomials in $t$ whose maximal minors have degree $q$. Results in [@Clark] imply that it suffices to consider the matrices ${\mathcal M}(t)$ in the introduction. The coefficients of these maximal minors define the [*Plücker embedding*]{} of $S^q_{p,m}$ into ${\mathbb P}(\wedge^pk^{m+p}\otimes k^{q+1})$; see [@Stromme; @Rosen94]. The [*quantum Grassmannian*]{} $K^q_{p,m}$ is the Zariski closure of $S^q_{p,m}$ in this Plücker embedding. It is an irreducible projective variety of dimension $mp+q(m+p)$. Its prime ideal is $\,{\rm kernel}(\varphi) \subset k[ {\mathcal C}^q_{p,m}]\,$ and its coordinate ring is our subalgebra $\,{\rm image}(\varphi)\subset k[X]$. The quantum Grassmannian $K^q_{p,m}$ is singular and it differs from other spaces used to study rational curves in Grassmann varieties (the quot scheme [@Stromme], the Kontsevich space of stable maps [@Kontsevich_Manin], or the set of autoregressive systems [@RR94]). Nevertheless, $K^q_{p,m}$ has been crucial in two important advances: in computing the intersection number $\,{\rm degree}(K^q_{p,m})\,$ in quantum cohomology [@RRW98], and in showing that this intersection problem can be fully solved over the real numbers [@Sottile_quantum]. Our result will give a new derivation of this intersection number. \[cor:counting\] [[@RRW98]]{} The degree of $\,K^q_{p,m} \,$ is the number of maximal chains in ${\mathcal C}^q_{p,m}$. [**Proof.** ]{} This follows immediately from Theorem 2 and Corollary \[Degree\]. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ This degree can also be computed in the small quantum cohomology ring of the Grassmannian [@Bertram]. Note that $\, {\rm degree} (K^1_{2,3}) = 55 $, by counting maximal chains in Figure \[fig:qorder\]. Ravi, Rosenthal, and Wang [@RRW98] were motivated by a problem in applied mathematics. The degree of $\,K^q_{p,m} \,$ is the number of dynamic feedback compensators that stabilize a certain linear system, in the sense of systems theory. This number can be described in classical projective geometry as follows. The Schubert subvariety of ${\it Grass}_pk^{m+p}$ consisting of $p$-planes meeting a fixed $m$-plane $L$ is a hyperplane section in the Plücker embedding of ${\it Grass}_pk^{m+p}$. Thus the set of maps $M\in S^q_{p,m}$ such that $M(t)$ meets $L$ non-trivially is a hyperplane section of $S^q_{p,m}$ in its Plücker embedding. Since $GL_{m+p}(k)$ acts transitively on ${\it Grass}_pk^{m+p}$, Kleiman’s Theorem on generic transversality implies the following statement when $k$ is algebraically closed of characteristic zero. Set $N:=mp+q(m+p)$ and suppose $t_1,\ldots,t_N\in{\mathbb P}^1$ are general points and $L_1,\ldots,L_N$ are general $m$-planes in $k^{m+p}$, then the degree of $K^q_{p,m}$ counts those maps $M$ for which $M(t_i)$ meets $L_i$ non-trivially for each $i=1,\ldots,N$. As to computing the desired maps $M$ numerically, we note that the sagbi basis in Theorem 1 and the Gröbner basis in Theorem 2 each lead to an [*optimal homotopy algorithm*]{} for finding these $\,{\rm degree}(K^q_{p,m})\,$ maps. These algorithms generalize the ones in [@HSS]. \[rem:flat\] The sagbi basis in Theorem 1 defines a flat deformation from the quantum Grassmannian $K^q_{p,m}$ to the projective toric variety $T^q_{p,m}$ associated with the poset ${\mathcal C}^q_{p,m}$. See [@CHV] for a precise algebraic discussion of such deformations, and see [@Sturmfels_GBCP Equation (11.9)] for the simplest example relevant to us, namely, $K^0_{2,3} = {\it Grass}_2 k^{5}$. The flat deformation is given algebraically by deleting all but the first two terms in the Gröbner basis elements $\, S(\gamma^{(c)},\delta^{(d)}) \,$ given in Theorem 2. Consider the deformation from $K^3_{3,3}$ to $T^3_{3,3}$. The incomparable pair $156^{(1)}$ and $234^{(2)}$ in ${\mathcal C}^3_{3,3}\,$ indexes the quadratic polynomial $\,\, S(156^{(1)},234^{(2)})\, = $ [ $$\label{BigSyzygy} \begin{array}{c} \underline{ 156^{(1)}234^{(2)} -146^{(1)}235^{(2)} }\, +145^{(1)}236^{(2)} +136^{(1)}245^{(2)} -135^{(1)}246^{(2)}\\ +134^{(1)}256^{(2)} -126^{(1)}345^{(2)} +125^{(1)}346^{(2)} -124^{(1)}356^{(2)} +123^{(1)}456^{(2)}\\ -456^{(0)}123^{(3)} +356^{(0)}124^{(3)} -346^{(0)}125^{(3)} +345^{(0)}126^{(3)} -256^{(0)}134^{(3)} \\ +246^{(0)}135^{(3)} -245^{(0)}136^{(3)} -236^{(0)}145^{(3)} +235^{(0)}146^{(3)} -234^{(0)}156^{(3)} \\ +2{\cdot}156^{(0)}234^{(3)} -2{\cdot}146^{(0)}235^{(3)} +2{\cdot}145^{(0)}236^{(3)} +2{\cdot}136^{(0)}245^{(3)} -2{\cdot}135^{(0)}246^{(3)}\,\\ +2{\cdot}134^{(0)}256^{(3)} -2{\cdot}126^{(0)}345^{(3)} +2{\cdot}125^{(0)}346^{(3)} -2{\cdot}124^{(0)}356^{(3)} +2{\cdot}123^{(0)}456^{(3)}, \end{array}$$ ]{} which vanishes on the quantum Grassmannian $K^3_{3,3}$. The underlined leading binomial vanishes on the toric variety $T^3_{3,3}$, by Proposition \[lem:toric\]. Our main technical problem, to be solved in the next section, is the reconstruction of quadrics such as (\[BigSyzygy\]) from their leading binomial. A key tool in proving Theorems 1 and 2 is the Schubert decomposition of the quantum Grassmannian $K^q_{p,m}$ indexed by ${\mathcal C}^q_{p,m}$. For $\alpha^{(a)}\in{\mathcal C}^q_{p,m}$, the [*quantum Schubert variety*]{} is $$Z_{\alpha^{(a)}}\quad :=\quad \bigl\{ \,({\gamma^{(c)}})\in K^q_{p,m} \,\mid\, {\gamma^{(c)}}=0 \, \mbox{ if } \, \gamma^{(c)}\not\leq \alpha^{(a)} \bigr\}\ .$$ More generally, for $\beta^{(b)}\leq \alpha^{(a)}$ in ${\mathcal C}^q_{p,m}$, we define the [*skew quantum Schubert variety*]{} $$Z_{\alpha^{(a)}/\beta^{(b)}}\quad :=\quad \bigl\{\, ({\gamma^{(c)}})\in K^q_{p,m} \,\mid \, {\gamma^{(c)}} = 0 \, \mbox{ if } \, \gamma^{(c)}\not\in [\beta^{(b)},\alpha^{(a)}] \bigr\}\ .$$ Among the quantum Schubert varieties of $K^q_{p,m}$ are the $K^d_{p,m}$ for $d<q$; namely, if $\delta^{(d)}$ is the supremum of ${\mathcal C}^d_{p,m}$, then $K^d_{p,m}=Z_{\delta^{(d)}}$. This allows us to deduce assertions about the general quantum Grassmannian $K^q_{p,m}$ from results about quantum Schubert varieties of $K^{pn}_{p,m}$. The quantum Schubert varieties and skew quantum Schubert varieties have rational parameterizations which are constructed as follows. Let $\alpha^{(a)}\in{\mathcal C}^{pn}_{p,m}$ and write $a=ps+r$ with integers $p>r\geq 0$. We define the matrix ${\mathcal M}_{\alpha^{(a)}}(t)$ to be the specialization of ${\mathcal M}(t)$ where $$x_{i,j}^{(l)}\ =\ 0\quad\mbox{if }\quad\left\{ \begin{array}{ll} ( l>s+1 \ \mbox{ and } \ i\leq r ) &\mbox{ or } \ \ (l=s+1\mbox{ and }j>\alpha_{r+1-i}) \ \ \mbox{or} \\ ( l>s \ \mbox{ and } \ i>r ) &\mbox{ or } \ \ (l=s\mbox{ and }j>\alpha_{p+r+1-i})\,. \end{array}\right.$$ Here we use the conventions $\alpha_\nu = 0$ if $\nu \leq 0\, $ and $\,\alpha_\nu = +\infty$ if $\nu > p$. For example, $${\mathcal M}_{235^{(2)}}(t) \quad = \quad \, \left[\begin{array}{cccccc} x_{1,1}^{(0)}+x_{1,1}^{(1)}\cdot t&x_{1,2}^{(0)}+x_{1,2}^{(1)}\cdot t &x_{1,3}^{(0)}+x_{1,3}^{(1)}\cdot t&x_{1,4}^{(0)} &\ x_{1,5}^{(0)}&\ x_{1,6}^{(0)}\\ \rule{0pt}{17pt} x_{2,1}^{(0)}+x_{2,1}^{(1)}\cdot t&x_{2,2}^{(0)}+x_{2,2}^{(1)}\cdot t &x_{2,3}^{(0)}\ \ \ &\ x_{2,4}^{(0)}\ &\ x_{2,5}^{(0)} &\ x_{2,6}^{(0)} \\ \rule{0pt}{17pt} x_{3,1}^{(0)}\ \ \ &x_{3,2}^{(0)}\ \ \ & x_{3,3}^{(0)}\ \ \ &\ x_{3,4}^{(0)}\ &\ x_{3,5}^{(0)} &\ \ 0 \rule{0pt}{16pt} \end{array}\right].$$ If we specialize the variables $x^{(l)}_{i,j}$ in ${\mathcal M}_{\alpha^{(a)}}(t)$ to field elements in $k$ in such a way that the resulting matrix over $k(t)$ has maximal row rank, then that matrix defines a map from $k$ to ${\it Grass}_pk^{m+p}$. If we extend this to ${\mathbb P}^1$, we obtain a map in $Z_{\alpha^{(a)}}$. Proposition \[ItIsDominant\] below implies that such maps constitute a dense subset of $Z_{\alpha^{(a)}}$. This means that the coefficients with respect to $t$ of the maximal minors of ${\mathcal M}_{\alpha^{(a)}}(t)$ give a rational parameterization of $Z_{\alpha^{(a)}}$. This construction extends to skew quantum Schubert varieties as follows. Given $\beta^{(b)}\leq\alpha^{(a)}$, write $b=ps+r$ with integers $p>r\geq 0$ and define the matrix ${\mathcal M}_{\alpha^{(a)}/\beta^{(b)}}(t)$ to be the specialization of ${\mathcal M}_{\alpha^{(a)}}(t)$ where $$x_{i,j}^{(l)}\ =\ 0\quad\mbox{if }\quad\left\{ \begin{array}{ll} ( l<s+1 \ \mbox{ and }\ i\leq r ) &\mbox{ or } \ \ ( l=s+1\mbox{ and }j< \beta_{r+1-i}) \ \ \mbox{ or } \\ ( l<s \ \mbox{ and }\ i>r ) & \mbox{ or } \ \ ( l=s\mbox{ and }j<\beta_{p+r+1-i}) \end{array}\right.\,.$$ The matrix ${\mathcal M}_{\alpha^{(a)}/\beta^{(b)}}(t)$ gives a rational map into $Z_{\alpha^{(a)}/\beta^{(b)}}$, which is described algebraically as follows. We define $\varphi_{\alpha^{(a)}}$ and $\varphi_{\alpha^{(a)}/\beta^{(b)}}$ to be the composition of the map $\,\varphi : k[{\mathcal C}^{np}_{p,m}] \rightarrow k[X] \,$ with the specializations to ${\mathcal M}_{\alpha^{(a)}}(t)$ and ${\mathcal M}_{\alpha^{(a)}/\beta^{(b)}}(t)$ respectively. We claim that these matrices parameterize dense subsets of the (skew) quantum Schubert varieties. \[ItIsDominant\] The kernel of $\varphi_{\alpha^{(a)}}$ is the homogeneous ideal of the quantum Schubert variety $Z_{\alpha^{(a)}}$. Likewise, the kernel of $\varphi_{\alpha^{(a)}/\beta^{(b)}}$ is the homogeneous ideal of the skew quantum Schubert variety $Z_{\alpha^{(a)}/\beta^{(b)}}$. In particular, the varieties $Z_{\alpha^{(a)}}$ and $Z_{\alpha^{(a)}/\beta^{(b)}}$ are irreducible. We postpone the proof of this proposition until the next section. Here is an example which illustrates the parameterization of skew quantum Schubert varieties for $p = m = 3$: $${\mathcal M}_{235^{(2)}/146^{(1)}}(t) \quad = \quad \left[\begin{array}{cccccc} x_{1,1}^{(1)}\cdot t&x_{1,2}^{(1)}\cdot t &x_{1,3}^{(1)}\cdot t&0&0&0\\ \rule{0pt}{17pt} x_{2,1}^{(1)}\cdot t&x_{2,2}^{(1)}\cdot t &0&0&0 &x_{2,6}^{(0)} \\ \rule{0pt}{17pt} 0&0&0&x_{3,4}^{(0)}&x_{3,5}^{(0)}& 0 \rule{0pt}{16pt} \end{array}\right]\,.$$ We evaluate the $3 \times 3$-minors of this matrix to find the $k$-algebra homomorphism $\,\varphi_{{235^{(2)}/146^{(1)}}} $. It takes polynomials in $12$ variables $\,\gamma^{(c)}\,$ to polynomials in $8$ variables $x^{(l)}_{i,j}$ as follows: $$\begin{array}{c} 146^{(1)} \mapsto -x^{(1)}_{1,1} x^{(0)}_{2,6} x^{(0)}_{3,4} \,\,, \quad 156^{(1)} \mapsto -x^{(1)}_{1,1} x^{(0)}_{2,6} x^{(0)}_{3,5} \,\, ,\quad 246^{(1)} \mapsto -x^{(1)}_{1,2} x^{(0)}_{2,6} x^{(0)}_{3,4} \,, \\\rule{0pt}{15pt} 256^{(1)} \mapsto -x^{(1)}_{1,2} x^{(0)}_{2,6} x^{(0)}_{3,5} \,\, ,\quad 346^{(1)} \mapsto -x^{(1)}_{1,3} x^{(0)}_{2,6} x^{(0)}_{3,4} \,\, ,\quad 356^{(1)} \mapsto -x^{(1)}_{1,3} x^{(0)}_{2,6} x^{(0)}_{3,5} \, , \\\rule{0pt}{15pt} 124^{(2)} \mapsto x^{(1)}_{1,1} x^{(1)}_{2,2} x^{(0)}_{3,4} - x^{(1)}_{2,1} x^{(1)}_{1,2} x^{(0)}_{3,4} \, , \quad 125^{(2)} \mapsto x^{(1)}_{1,1} x^{(1)}_{2,2} x^{(0)}_{3,5} - x^{(1)}_{2,1} x^{(1)}_{1,2} x^{(0)}_{3,5}\, , \\\rule{0pt}{15pt} 134^{(2)} \mapsto -x^{(1)}_{2,1} x^{(1)}_{1,3} x^{(0)}_{3,4} \,\, ,\qquad 135^{(2)} \mapsto -x^{(1)}_{2,1} x^{(1)}_{1,3} x^{(0)}_{3,5} \,, \\\rule{0pt}{15pt} 234^{(2)} \mapsto -x^{(1)}_{2,2} x^{(1)}_{1,3} x^{(0)}_{3,4} \,\, ,\qquad 235^{(2)} \mapsto -x^{(1)}_{2,2} x^{(1)}_{1,3} x^{(0)}_{3,5} .\end{array}\vspace{-4pt}$$ The $12$ variables $\gamma^{(c)}$ appearing on the left sides above are precisely the elements in the interval $\,\bigl[146^{(1)}, 235^{(2)} \bigr]\,$ of the distributive lattice ${\mathcal C}_{3,3}$. There are $18$ incomparable pairs in this interval, each giving a quadratic generator for the kernel of $\,\varphi_{{235^{(2)}/146^{(1)}}} $. This set of $18$ quadrics consists of $14$ binomials and four trinomials, and it equals the reduced Gröbner basis with respect to $\prec$. For example, one of the 14 binomials in this Gröbner basis is the underlined leading binomial of $\,\, S(156^{(1)},234^{(2)})\, $ in (\[BigSyzygy\]), and one of the four trinomials is $$\underline{ 346^{(1)} \cdot 125^{(2)} } \,\, - \,\, 246^{(1)} \cdot 135^{(2)} \,\, + \,\, 146^{(1)} \cdot 235^{(2)}.$$ The underlined term is an incomparable pair in $\bigl[146^{(1)}, 235^{(2)} \bigr]$, while the other two monomials are comparable pairs. Erasing the third term gives a binomial as in Proposition \[lem:toric\]. Construction of Straightening Syzygies {#sec:syzygy} ====================================== The following theorem is the technical heart of this paper. All three of Theorem \[issagbi\], Theorem \[thm:gbasis\], and Proposition \[ItIsDominant\] will be derived from Theorem \[thm:syzygy\] in the end of this section. \[thm:syzygy\] Let $\gamma^{(c)},\delta^{(d)}$ be a pair of incomparable variables in the poset ${\mathcal C}^{np}_{p,m}$. There is a quadric $S(\gamma^{(c)},\delta^{(d)})$ in the kernel of $\varphi : k[{\mathcal C}^{np}_{p,m}] \rightarrow k[X]$ whose first two monomials are $$\gamma^{(c)}\cdot\delta^{(d)}\ -\ (\gamma^{(c)}\vee\delta^{(d)}) \cdot (\gamma^{(c)}\wedge\delta^{(d)}).$$ Moreover, if $\lambda\beta^{(b)}\alpha^{(a)}$ is any non-initial monomial in $S(\gamma^{(c)},\delta^{(d)})$, then $\gamma^{(c)},\delta^{(d)}\in[\beta^{(b)},\alpha^{(a)}]$. The pair $ \beta^{(b)}\alpha^{(a)}$ in the second assertion is necessarily standard, i.e. $\beta^{(b)}<\alpha^{(a)}$. The quadrics $S(\gamma^{(c)},\delta^{(d)})$ are not constructed explicitly, but rather through an iterative procedure modeled on the [*subduction algorithm*]{} in [image]{}$(\varphi)$. A main idea is to utilize the well-known subduction process [@Sturmfels_invariant Algorithm 3.2.6] modulo the $p \times p$-minors of a generic $p \times N$-matrix. Set $N:=(n+1)(m+p)$. Let ${\mathcal N}$ be the $p\times N$-matrix whose $i,j$th entry is $x^{(l)}_{i,r}$, where $j= (m+p)l+r$ with $1\leq r\leq m+p$. If ${\mathcal N}_l$ is the submatrix of ${\mathcal N}$ consisting of the entries $x^{(l)}_{i,j}$, then ${\mathcal N}$ is the concatenation of ${\mathcal N}_0,{\mathcal N}_1,\ldots,{\mathcal N}_n$ and ${\mathcal M}(t)= {\mathcal N}_0+t{\mathcal N}_1+\cdots+t^n{\mathcal N}_n$. Sequences $\,J:j_1<\cdots<j_p\in\binom{[ N]}{p}\,$ are regarded as variables. We write $\phi(J)$ for the $J$th maximal minor of ${\mathcal N}$. [*Young’s poset*]{} on sequences $J$ is given by componentwise comparison and is graded via $\,|J| := \sum_i (j_i-i)$. The coefficient $\varphi(\alpha^{(a)})$ of $t^a$ in the $\alpha$th maximal minor of ${\mathcal M}(t)$ is an alternating sum of maximal minors of ${\mathcal N}$. The exact formula is $$\label{eq:phi_expand} \varphi(\alpha^{(a)})\quad \,\,\, =\quad \sum_{\stackrel{\mbox{\scriptsize $|J|=|\alpha^{(a)}|$}} {J\equiv\alpha\,\bmod{(m+p)}}} \!\! \epsilon_J \cdot \phi(J),$$ where $\epsilon_J$ is the sign of the permutation that orders the following sequence: $$j_1\bmod(m+p),\ j_2\bmod(m+p),\ \ldots,\ j_p\bmod(m+p).$$ The polynomial rings $k[{\mathcal C}^{np}_{p,m}]$ and $k[\binom{[N]}{p}]$ are graded with $\deg\alpha^{(a)}=|\alpha^{(a)}|$ and $\deg J=|J|$. Consider the degree-preserving $k$-algebra homomorphism $\pi:k[{\mathcal C}^{np}_{p,m}]\rightarrow k[\binom{[N]}{p}]$ defined by $$\label{eq:pidef} \pi(\alpha^{(a)})\quad \,\, \, =\quad \sum_{\stackrel{\mbox{\scriptsize $|J|=|\alpha^{(a)}|$}} {J\equiv\alpha\,\bmod{(m+p)}}} \!\! \epsilon_J \cdot J\,.$$ Lexicographic order on the sequences $J\in\binom{[ N]}{p}$ gives a linear extension of Young’s poset. In this ordering, the initial term of (\[eq:pidef\]) is the sequence $J(\alpha^{(a)})$ defined in (\[eq:seq\_def\]). This sequence is characterized by $\, {\rm in}_\prec \,\varphi(\alpha^{(a)}) \, = \,{\rm in}_\prec \, \phi(J(\alpha^{(a)}))$. It can be checked that all other terms $ \epsilon_J \cdot J \,$ appearing in (\[eq:pidef\]) satisfy $\, J_1 < J(\alpha^{(a)})_1 \,$ and $\, J_p-J_1 \ >\ m+p $. For example, for $m = 4$, $$\pi(235^{(2)}) \quad = \quad \underline{ (5, 9, 10)} - (3, 9, 12) + (3, 5, 16) + (2, 10, 12) - (2, 5, 17) + (2,3,19),$$ $$\mbox{and} \qquad {\rm in}_\prec \bigl( \varphi( \,235^{(2)} \,) \bigr) \quad = \quad {\rm in}_\prec\bigl(\phi( 5,9,10) \bigr) \quad = \quad x_{3,5}^{(0)} x_{1,3}^{(1)} x_{2,2}^{(1)} . \qquad$$ For $J\in\binom{[ N]}{p}$, let ${\mathcal N}_J$ be the specialization of ${\mathcal N}$ where in each row $i$, all entries in columns greater than $j_i$ are set to zero. Under the identification of ${\mathcal N}$ with ${\mathcal M}(t)$, we have ${\mathcal N}_{J(\alpha^{(a)})}={\mathcal M}_{\alpha^{(a)}}$. Let $\phi_J : k [\binom{[N]}{p}] \rightarrow k[X] $ denote the $k$-algebra homomorphism which maps the formal variable $I$ to the $I$th maximal minor of ${\mathcal N}_J$. Then $\phi_J(I)$ vanishes unless $I\leq J$. In particular, if $|I|=|J|$, then $\phi_J(I)$ vanishes unless $I=J$, and in that case, it is just the product of the last non-zero variables in each row of ${\mathcal N}_J$. From this it follows that $$\label{SomeIdentities} \begin{array}{rcl} & \varphi_{\alpha^{(a)}} \quad = \quad \phi_{J(\alpha^{(a)})} \circ \pi \\\rule{0pt}{15pt} & \varphi_{\alpha^{(a)}}(\alpha^{(a)}) \,\, =\,\, \phi_{J(\alpha^{(a)})}(J(\alpha^{(a)})) \,\, = \,\, {\rm in}_\prec \, \varphi(\alpha^{(a)}) \,\, = \,\, \psi(\alpha^{(a)}). \end{array}$$ In the Plücker embedding of [*Grass*]{}$_pk^{ N}$ into ${\mathbb P}(\wedge^p k^{N})$, the Schubert variety indexed by $J$ is $$\Omega_J\quad :=\quad \{y=(y_I)\in {\it Grass}_pk^{ N}\mid y_I=0\mbox{ if }I\not\leq J \}\ .$$ The homogeneous ideal ${\mathcal I}(\Omega_J)$ which defines this Schubert variety is precisely the kernel of $\phi_J$. The following identity of ideals in $k[\binom{[N]}{p}]$ follows from the classical Plücker relations: \[prop:Schubert-ideal\] For any $J\in\binom{[ N]}{p}$ we have $$\bigcap_{I<J}{\mathcal I}(\Omega_I)\quad =\quad {\mathcal I}(\Omega_J) \, + \, \left\langle\, J \,\right\rangle\,.$$ The map $\pi:k[{\mathcal C}^{np}_{p,m}]\rightarrow k[\binom{[N]}{p}]$ induces a birational isomorphism $\pi^*:{\it Grass}_pk^{ N}\dashrightarrow K^{np}_{p,m}$. From the identification of ${\mathcal M}_{\alpha^{(a)}}(t)$ with ${\mathcal N}_{J(\alpha^{(a)})}$ and Proposition \[ItIsDominant\], we will see that $\pi^*(\Omega_{J(\alpha^{(a)})})$ is a dense subset of $\, Z_{\alpha^{(a)}}$. We also consider the image under $\pi^*$ of the Schubert varieties $\Omega_J$ for $\, J<J(\alpha^{(a)})$. \[prop:O\_J-image\] If $J<J(\alpha^{(a)})$, then $$\pi^*(\Omega_J)\quad \subset\quad \bigcup_{\beta^{(b)}<\alpha^{(a)}} Z_{\beta^{(b)}}\,.$$ [**Proof.** ]{} The inclusion $\Omega_J\subset\Omega_{J(\alpha^{(a)})}$ implies $\pi^*(\Omega_J)\subset Z_{\alpha^{(a)}}$. Since $\varphi_{\alpha^{(a)}}(\alpha^{(a)})$ is the product of leading entries in the rows of ${\mathcal N}_{J(\alpha^{(a)})}$, it follows that $\varphi_{\alpha^{(a)}}(\alpha^{(a)})$ vanishes under the specialization to ${\mathcal N}_J$, and hence $\pi(\alpha^{(a)})$ vanishes on $\Omega_J$. This implies our claim because $\bigcup_{\beta^{(b)}<\alpha^{(a)}} Z_{\beta^{(b)}}$ is defined as a subvariety of $ Z_{\beta^{(b)}}$ by the vanishing of $\alpha^{(a)}$. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ For $L<J$ in Young’s poset, define ${\mathcal N}_{J/L}$ to be the specialization of ${\mathcal N}$ where in the $i$th row, only the entries in columns $l_i,l_i+1,\ldots,j_i$ are non-zero. Then ${\mathcal M}_{\alpha^{(a)}/\beta^{(b)}}(t)$ is the specialization of ${\mathcal M}(t)$ corresponding to ${\mathcal N}_{J(\alpha^{(a)})/J(\beta^{(b)})}(t)$. Define the $k$-algebra homomorphism $\,\phi_{J/I} : k [\binom{[N]}{p}] \rightarrow k[X] \,$ by evaluating the appropriate minors on ${\mathcal N}_{J/L}$. We observe that $$\label{eq:initial} \begin{array}{rcl} \varphi_{\alpha^{(a)}/\beta^{(b)}}(\alpha^{(a)}) &=& {\rm in}_\prec \, \varphi(\alpha^{(a)}) \quad = \quad \phi_{J(\alpha^{(a)})/J(\beta^{(b)})}\bigl( J(\alpha^{(a)})\bigr), \\\rule{0pt}{16pt} \varphi_{\alpha^{(a)}/\beta^{(b)}}(\beta^{(b)}) &=& {\rm in}_\prec \, \varphi(\beta^{(b)}) \quad = \quad \phi_{J(\alpha^{(a)})/J(\beta^{(b)})}\bigl( J(\beta^{(b)})\bigr). \end{array}$$ The following lemma is very useful in our proof of Theorem \[thm:syzygy\]. \[lem:factor\] Fix $\alpha^{(a)} \in {\mathcal C}^{np}_{p,m}$ and let $f\in k[{\mathcal C}^{np}_{p,m}]$ be a quadratic form of degree $d$. 1. Suppose that $\varphi_{\beta^{(b)}}(f)=0$ for all $\beta^{(b)}<\alpha^{(a)}$. Then there exist constants $\lambda_J\in k$ with $$\varphi_{\alpha^{(a)}}(f)\quad =\quad \varphi_{\alpha^{(a)}}(\alpha^{(a)}) \,\, \cdot \!\!\!\! \sum_{\stackrel{\mbox{\scriptsize $J \in \binom{[ N]}{p}$}} {|J|+|\alpha^{(a)}|=d}} \lambda_J \cdot \phi_{J(\alpha^{(a)})}(J)\,.$$ 2. Suppose $\beta^{(b)}<\alpha^{(a)}$ and $\varphi_{\alpha^{(a)}/\gamma^{(c)}}(f)=0$ for all $\beta^{(b)}<\gamma^{(c)}\leq\alpha^{(a)}$. For some $\lambda_J\in k$, $$\varphi_{\alpha^{(a)}/\beta^{(b)}}(f)\quad =\quad \varphi_{\alpha^{(a)}/\beta^{(b)}}(\beta^{(b)}) \, \cdot \!\!\!\! \sum_{\stackrel{\mbox{\scriptsize $J \in \binom{[ N]}{p}$}} {|J|+|\beta^{(b)}|=d}} \lambda_J\cdot \phi_{J(\alpha^{(a)})/J(\beta^{(b)})}(J)\,.$$ [**Proof.** ]{} We only prove part 1. The hypothesis states that $\, \phi_{J(\alpha^{(a)})}(\pi(f)) \, = \, \varphi_{\alpha^{(a)}}(f) \, $ vanishes on all matrices ${\mathcal N}_{J(\beta^{(b)})}$ for $\beta^{(b)}<\alpha^{(a)}$. Proposition \[prop:O\_J-image\] implies that $\pi(f)$ vanishes on all Schubert varieties $\Omega_J$ with $J<J(\alpha^{(a)})$. But then, using Proposition \[prop:Schubert-ideal\], $$\pi(f)\quad \in\quad \bigcap_{J<J(\alpha^{(a)})}{\mathcal I}(\Omega_J)\quad =\quad {\mathcal I}(\Omega_{J(\alpha^{(a)})}) \, + \, \left\langle \, J(\alpha^{(a)}) \, \right\rangle.$$ This means $\,\pi(f)=g + J(\alpha^{(a)})\cdot h$, where $g\in {\mathcal I}(\Omega_{J(\alpha^{(a)})})= {\rm ker} \bigl(\phi_{J(\alpha^{(a)})} \bigr) $ and $h\in k[\binom{[N]}{p}]$ is a linear form of degree $d-|\alpha^{(a)}|$. Such a linear form can be written as follows $$h \, \quad =\quad \sum_{|J|+|\alpha^{(a)}|=d} \! \! \lambda_J\,J\,.$$ By applying the map $\, \phi_{J(\alpha^{(a)})} \,$ to both sides of the equation $\,\pi(f)=g + J(\alpha^{(a)})\cdot h$, we obtain the first assertion of Lemma \[lem:factor\]. Part 2 is proved by similar arguments. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ Our proof of Theorem \[thm:syzygy\] will show that the sums in Lemma \[lem:factor\] are actually sums of terms of the form $\,\lambda_{J(\delta^{(d)})}\cdot\varphi_{\alpha^{(a)}}(\delta^{(d)})\,$ and $\,\lambda_{J(\delta^{(d)})}\cdot \varphi_{\alpha^{(a)}/\beta^{(b)}}(\delta^{(d)}) \,$ respectively. The next lemma provides the initial step in our inductive proof of Theorem \[thm:syzygy\]. \[lem:initial\] Let $\gamma^{(c)}$ and $\delta^{(d)}$ be incomparable variables in the poset ${\mathcal C}^{np}_{p,m}$ and set $\alpha^{(a)}:=\gamma^{(c)}\vee\delta^{(d)}$. Then $ \varphi_{\alpha^{(a)}}(\gamma^{(c)}\cdot\delta^{(d)}\ -\ \gamma^{(c)}\vee\delta^{(d)}\cdot\gamma^{(c)}\wedge\delta^{(d)}) =0$. [**Proof.** ]{} We prove the lemma by inductively showing that, for each $\,\beta^{(b)}\leq\alpha^{(a)}$, $$\label{eq:lt} \varphi_{\alpha^{(a)}/\beta^{(b)}} \bigl( \, \gamma^{(c)}\cdot\delta^{(d)}\ -\ \gamma^{(c)}\vee\delta^{(d)}\cdot\gamma^{(c)}\wedge\delta^{(d)} \,\bigr) \quad = \quad 0.$$ If $\beta^{(b)}\not\leq\gamma^{(c)}\wedge\delta^{(d)}$, then $\varphi_{\alpha^{(a)}/\beta^{(b)}}(\gamma^{(c)}\wedge\delta^{(d)})$ vanishes, and either $\varphi_{\alpha^{(a)}/\beta^{(b)}}(\gamma^{(c)})$ vanishes or $\varphi_{\alpha^{(a)}/\beta^{(b)}}(\delta^{(d)})$ vanishes. This implies that (\[eq:lt\]) holds. Next suppose $\beta^{(b)}=\gamma^{(c)}\wedge\delta^{(d)}$. We claim that $\varphi_{\alpha^{(a)}/\beta^{(b)}}$ maps each variable appearing in (\[eq:lt\]) to its initial term in $k[X]$. In view of Proposition \[lem:toric\], this claim implies (\[eq:lt\]). To establish this claim, we need only show that $\varphi_{\alpha^{(a)}/\beta^{(b)}}(\gamma^{(c)})= {\rm in}_\prec\varphi(\gamma^{(c)})$, as the case for $\delta^{(d)}$ is similar and that of the other terms follow from (\[eq:initial\]). Consider the expansion of $\varphi_{\alpha^{(a)}/\beta^{(b)}}(\gamma^{(c)})$ in terms of the minors $\phi(J)$ of ${\mathcal N}_{J(\alpha^{(a)})/J(\beta^{(b)})}$. First observe that the submatrix given by the columns from $J(\gamma^{(c)})$ is block anti-diagonal, with each block either upper or lower triangular along its anti-diagonal. This is because for each $i$, $J(\gamma^{(c)})_i$ is either $J(\beta^{(b)})_i$ or $J(\alpha^{(a)})_i$, and the non-zero entries in the $i$th row of ${\mathcal N}_{J(\alpha^{(a)})/J(\beta^{(b)})}$ lie between these two numbers. Thus the contribution of term $J(\gamma^{(c)})$ to $\varphi_{\alpha^{(a)}/\beta^{(b)}}(\gamma^{(c)})$ is simply ${\rm in}_\prec \, \varphi(\gamma^{(c)})$. We claim there are no other terms. If there were another term indexed by $L$, then the $L$th maximal minor of ${\mathcal N}_{J(\alpha^{(a)})/J(\beta^{(b)})}$ would be non-zero, and so $J(\beta^{(b)})\leq L\leq J(\alpha^{(a)})$. Thus $$J(\beta^{(b)})\quad =\quad J(\gamma^{(c)})\wedge J(\delta^{(d)}) \quad \leq\quad L\wedge J(\delta^{(d)}).$$ Comparing the first components of these sequences gives $\min\{J(\gamma^{(c)})_1,J(\delta^{(d)})_1\}\leq L_1$. Since $L_1<J(\gamma^{(c)})_1$, this implies $J(\delta^{(d)})_1\leq L_1$. Similarly, using $J(\alpha^{(a)})\geq L\vee J(\delta^{(d)})$, we see that $J(\delta^{(d)})_p\geq L_p$. Lastly, as $L$ is a summand in $\pi(\gamma^{(c)})$ and $L\neq J(\gamma^{(c)})$, we have $L_p-L_1> m+p$ and thus $$m+p\ \geq\ J(\delta^{(d)})_p-J(\delta^{(d)})_1\ \geq\ L_p-L_1\ >\ m+p,$$ a contradiction, which proves the claim. Thus (\[eq:lt\]) holds for $\beta^{(b)}=\gamma^{(c)}\wedge\delta^{(d)}$. Finally, let $\zeta^{(z)}<\gamma^{(c)}\wedge\delta^{(d)}$ and suppose that (\[eq:lt\]) holds for all $\beta^{(b)}$ with $\zeta^{(z)}<\beta^{(b)}\leq \alpha^{(a)}$. Then by Lemma \[lem:factor\], $$\varphi_{\alpha^{(a)}/\zeta^{(z)}}\bigl(\gamma^{(c)}\cdot\delta^{(d)}\ -\ \gamma^{(c)}\vee\delta^{(d)}\cdot\gamma^{(c)}\wedge\delta^{(d)} \bigr) \quad =\quad \varphi_{\alpha^{(a)}/\zeta^{(z)}}(\zeta^{(z)})\cdot \sum_{J}\lambda_J \cdot \phi_{J(\alpha^{(a)})/J(\zeta^{(z)})}(J)\,,$$ the sum over sequences $J$ of rank $|J|=|\gamma^{(c)}|+|\delta^{(d)}|-|\zeta^{(z)}|$. But this exceeds the rank of $\alpha^{(a)}$, since $\zeta^{(z)}<\gamma^{(c)}\wedge\delta^{(d)}$ and $|\alpha^{(a)}|+|\gamma^{(c)}\wedge\delta^{(d)}| =|\gamma^{(c)}|+|\delta^{(d)}|$. Thus the sum vanishes and so (\[eq:lt\]) holds for all $\beta^{(b)}\leq\alpha^{(a)}$, which proves the lemma. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ [**Proof of Theorem \[thm:syzygy\].** ]{} Let $\gamma^{(c)}$ and $\delta^{(d)}$ be incomparable variables in the poset ${\mathcal C}^{np}_{p,m}$. For each $\alpha^{(a)}\in{\mathcal C}^{np}_{p,m}$ we inductively construct quadratic polynomials $S_{\alpha^{(a)}}(\gamma^{(c)},\delta^{(d)})\in k[\beta^{(s)}\mid\beta^{(s)}\leq \alpha^{(a)}]$, and then show $S_{\alpha^{(a)}}(\gamma^{(c)},\delta^{(d)})$ is in the kernel of the map $\varphi_{\alpha^{(a)}}$. The case when $\alpha^{(a)}$ is the top element in the poset ${\mathcal C}^{np}_{p,m}$ proves the theorem. These polynomials have the following restriction property: If $\beta^{(b)}<\alpha^{(a)}$, then $S_{\beta^{(b)}}(\gamma^{(c)},\delta^{(d)})$ is the image of $S_{\alpha^{(a)}}(\gamma^{(c)},\delta^{(d)})$ under the map which sets variables $\zeta^{(z)}\not\leq \beta^{(b)}$ to zero. They also have the further homogeneity properties that each non-zero term $\lambda\zeta^{(z)}\beta^{(b)}$ must have $z+b=c+d$ and satisfy the multiset equality $\beta\cup\zeta=\gamma\cup\delta$, and if it is not the initial term, then $\gamma^{(c)},\delta^{(d)}\in[\zeta^{(z)},\beta^{(b)}]$. For $\alpha^{(a)}\not\geq \gamma^{(c)}\vee\delta^{(d)}$, set $S_{\alpha^{(a)}}(\gamma^{(c)},\delta^{(d)}):=0$ and if $\alpha^{(a)}=\gamma^{(c)}\vee\delta^{(d)}$, then set $$S_{\alpha^{(a)}}(\gamma^{(c)},\delta^{(d)})\quad:=\quad \gamma^{(c)}\cdot\delta^{(d)}\ -\ \gamma^{(c)}\vee\delta^{(d)}\cdot\gamma^{(c)}\wedge\delta^{(d)}.$$ These polynomials have the restriction and homogeneity properties, and, for $\alpha^{(a)}\not> \gamma^{(c)}\vee\delta^{(d)}$, we have $\varphi_{\alpha^{(a)}}(S_{\alpha^{(a)}}(\gamma^{(c)},\delta^{(d)}))=0$, by Lemma \[lem:initial\]. Let $\alpha^{(a)}>\gamma^{(c)}\vee\delta^{(d)}$ and suppose we have constructed $S_{\beta^{(b)}}(\gamma^{(c)},\delta^{(d)})$ for each $\beta^{(b)}<\alpha^{(a)}$. By the restriction property, there is a polynomial $S'\in k[\beta^{(b)}\mid \beta^{(b)}<\alpha^{(a)}]$ which restricts to $S_{\beta^{(b)}}(\gamma^{(c)},\delta^{(d)})$ for each $\beta^{(b)}<\alpha^{(a)}$. Thus $\varphi_{\beta^{(b)}}(S')=0$ for all $\beta^{(b)}<\alpha^{(a)}$. Set $e:=|\gamma^{(c)}|+|\delta^{(d)}|$, the degree of $S'$. By Lemma \[lem:factor\], $$\label{eq:sprime} \varphi_{\alpha^{(a)}}(S')\quad =\quad \varphi_{\alpha^{(a)}}(\alpha^{(a)})\cdot \sum_{|J|+|\alpha^{(a)}|=e}\lambda_J \cdot\phi_{J(\alpha^{(a)})}(J)\,.$$ If we consider the columns of ${\mathcal M}_{\alpha^{(a)}}(t)$ involved in $\varphi_{\alpha^{(a)}}(S')$, we see that this sum is further restricted to those $J$ which satisfy the multiset equality $(\gamma \cup\delta)\setminus \alpha\equiv J\mod(m+p)$, with $J\bmod(m+p)$ consisting of distinct integers, and with $J\leq J(\alpha^{(a)})$. If there are no such $J$, then $\varphi_{\alpha^{(a)}}(S')=0$ and we set $S_{\alpha^{(a)}}(\gamma^{(c)},\delta^{(d)})=S'$. Otherwise, let $z:=c+d-a$ and $\zeta:=(\gamma\cup\delta)\setminus\alpha$. Then the summands in (\[eq:sprime\]) are among those $J$ which appear in $\pi(\zeta^{(z)})$ so we have $J(\zeta^{(z)})<J(\alpha^{(a)})$ and hence $\zeta^{(z)}<\alpha^{(a)}$. Observe that $\varphi_{\alpha^{(a)}/\zeta^{(z)}}(S')= \lambda_{J(\zeta^{(z)})} \cdot \varphi_{\alpha^{(a)}/\zeta^{(z)}}(\alpha^{(a)}\zeta^{(z)})$. Define $$S_{\alpha^{(a)}}(\gamma^{(c)},\delta^{(d)})\quad :=\quad S' - \lambda_{J(\zeta^{(z)})} \alpha^{(a)}\zeta^{(z)}.$$ We claim that if $\lambda_{J(\zeta^{(z)})}\neq 0$, then $\zeta^{(z)}\leq\gamma^{(c)},\delta^{(d)}$. If not, then every term of $S'$ contains a variable $\xi^{(x)}$ with $\zeta^{(z)}\not\leq\xi^{(x)}$, and so we must have $\varphi_{\alpha^{(a)}/\zeta^{(z)}}(S')=0$, a contradiction. We complete the proof of Theorem \[thm:syzygy\] by showing that for $\beta^{(b)}\leq \alpha^{(a)}$, $$\label{eq:indstep} \varphi_{\alpha^{(a)}/\beta^{(b)}}(S_{\alpha^{(a)}} (\gamma^{(c)},\delta^{(d)}))\quad =\quad 0.$$ If $\beta^{(b)}\not\leq \zeta^{(z)}$, then $\varphi_{\alpha^{(a)}/\beta^{(b)}}(\zeta^{(z)})=0$ and so $\varphi_{\alpha^{(a)}/\beta^{(b)}}(S_{\alpha^{(a)}} (\gamma^{(c)},\delta^{(d)}))=\varphi_{\alpha^{(a)}/\beta^{(b)}}(S')$, which is zero as $\phi_{J(\alpha^{(a)})/J(\beta^{(b)})}(J)=0$ for all $J$ which appear in $\pi(\zeta^{(z)})$. By the construction of $S_{\alpha^{(a)}}(\gamma^{(c)},\delta^{(d)}))$, we also have $\varphi_{\alpha^{(a)}/\zeta^{(z)}}(S_{\alpha^{(a)}} (\gamma^{(c)},\delta^{(d)}))=0$. Let $\xi^{(x)}<\zeta^{(z)}$ and suppose (\[eq:indstep\]) holds for all $\beta^{(b)}$ with $\xi^{(x)}<\beta^{(b)}$. Then by Lemma \[lem:factor\], $$\varphi_{\alpha^{(a)}/\xi^{(x)}}(S_{\alpha^{(a)}} (\gamma^{(c)},\delta^{(d)}))\quad =\quad \varphi_{\alpha^{(a)}/\xi^{(x)}}(\xi^{(x)})\cdot \sum_{|J|+|\xi^{(x)}|=e} \lambda_J \cdot \phi_{J(\alpha^{(a)})/J(\xi^{(x)})}(J).$$ Since $|J|=e-|\xi^{(x)}|>e-|\zeta^{(z)}|=|\alpha^{(a)}|$, each term in the right hand sum is zero. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ It is now straightforward to derive all our assertions that were left unproven so far. [**Proof of Theorem \[issagbi\].**]{} Theorem \[thm:syzygy\] together with Proposition \[lem:toric\] shows that the subduction criterion for sagbi bases (see e.g. [@CHV Proposition 1.1] or [@Sturmfels_GBCP Theorem 11.4]) is satisfied. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ [**Proof of Theorem \[thm:gbasis\].** ]{} A standard fact on sagbi bases, proved in [@CHV Corollary 2.2] or in [@Sturmfels_GBCP Corollary 11.6 (1)], states that the reduced Gröbner basis for the binomial ideal ${\rm kernel}(\psi)$ lifts to a reduced Gröbner basis for the non-binomial ideal ${\rm kernel}(\varphi)$. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ [**Proof of Proposition \[ItIsDominant\].** ]{} If $\beta^{(b)}$ is the minimal element in the poset ${\mathcal C}_{p,m}$, then $\varphi_{\alpha^{(a)}}$ and $\varphi_{\alpha^{(a)}/\beta^{(b)}}$ have the same kernel, as the varieties $Z_{\alpha^{(a)}/\beta^{(b)}}$ and $Z_{\alpha^{(a)}}$ are equal. Hence it suffices to prove the second statement about $\varphi_{\alpha^{(a)}/\beta^{(b)}}$. Clearly, the kernel of $\varphi_{\alpha^{(a)}/\beta^{(b)}}$ contains the homogeneous ideal of the skew quantum Schubert variety $Z_{\alpha^{(a)}/\beta^{(b)}}$. If this containment were proper, then we would also get proper containment at the level of initial ideals with respect to the induced partial term order, which was denoted by ${\mathcal A}^T \omega$ in [@Sturmfels_GBCP Chapter 11]. But that is impossible since every binomial relation on the monomials $\,{\rm in}_\prec \, \varphi_{\alpha^{(a)}/\beta^{(b)}}( \gamma^{(c)}) = {\rm in}_\prec \, \varphi ( \gamma^{(c)}) \,$ lifts to a polynomial which vanishes on $Z_{\alpha^{(a)}/\beta^{(b)}}$, as shown in the proof Theorem \[thm:syzygy\]. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ Applications and future directions ================================== We first summarize some algebraic consequences of our main results. \[ASL\] The coordinate ring of the quantum Grassmannian $K^q_{p,m}$ is an algebra with straightening law on the distributive lattice $C^q_{p,m}$. It has a presentation by a non-commutative Gröbner basis consisting of quadratic elements, and, in particular, it is a Koszul algebra. [**Proof.**]{} The first statement follows from Theorem \[issagbi\] and the form of the syzygies $S(\gamma^{(c)},\delta^{(d)})$ of Theorem \[thm:gbasis\]. For the second statement, consider the coordinate ring of $K^q_{p,m}$ as the quotient of the free associative algebra on $C^q_{p,m}$ modulo a two-sided ideal. By [@EPS Proposition 3.2] that two-sided ideal has a quadratic Gröbner basis, obtained from lifting the Gröbner basis in Theorem \[thm:gbasis\]. For the classical Grassmannian $(n=0)$ this result appeared in [@Gr_Hu]. The Koszul property is a well-known consequence of the existence of a quadratic Gröbner basis; see e.g. [@Gr_Hu Theorem 3] for the non-commutative version which is relevant here. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ \[C-MG\] The coordinate ring of $K^q_{p,m}$ is a normal Cohen-Macaulay and Gorenstein domain. It has rational singularities if ${\rm char}(k) = 0$ and it is $F$-rational if ${\rm char}(k) > 0$. [**Proof.** ]{} By Corollary \[ASL\] and the results of [@CHV], these properties of $K^q_{p,m}$ follow from the corresponding properties of the toric variety $T^q_{p,m}$. But these were established in [@Wagner], as $T^q_{p,m}$ is the toric variety associated to the distributive lattice ${\mathcal C}^q_{p,m}$. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ We remark that both Corollary \[C-MG\] and the analog of Corollary \[ASL\] (with the poset ${\mathcal C}^q_{p,m}$ replaced by the appropriate interval) hold for the skew quantum Schubert varieties. Our next application is the sagbi property of the row-consecutive $p \times p$-minors of a matrix of indeterminates. This result is non-trivial since the set of [*all*]{} $p\times p$-minors is not a sagbi basis in general [@Sturmfels_GBCP Example 11.3]. A finite sagbi basis for the algebra of all $p \times p$-minors was found by Bruns and Conca [@Bruns_Conca]. Let ${\mathcal L}$ be the $p(n+1)\times(m+p)$-matrix whose $i,j$th entry is $x^{(l)}_{r,j}$, where $i=pl+r$. This matrix is obtained from ${\mathcal N}$ by stacking the matrices ${\mathcal N}_0, \ldots, {\mathcal N}_n$. Let $\chi:k[{\mathcal C}^{np}_{p,m}]\rightarrow k[X]$ denote the $k$-algebra homomorphism which sends the variable $\alpha^{(a)}$ to the $\alpha$th maximal minor of the submatrix of ${\mathcal L}$ consisting of rows $a+1,a+2,\ldots,a+p$. Thus the collection of polynomials $\chi(\alpha^{(a)})$ are the row-consecutive $p\times p$-minors of ${\mathcal L}$. \[row-c-sagbi\] The set $\, \bigl\{ \chi(\alpha^{(a)}) \, : \, \alpha^{(a)} \in {\mathcal C}^{np}_{p,m} \bigr\} \,$ of row-consecutive $p \times p$-minors of a generic matrix is a sagbi basis with respect to the degree reverse lexicographic term order $\prec$ on $k[X]$. Brain Taylor has pointed out this may also be deduced from Proposition 2.7.3 of his Ph.D. Thesis [@BDT_thesis]. [**Proof.** ]{} Let $\omega$ be the weight on the variables in $k[X]$ defined by $\omega(x^{(l)}_{i,j}):= -(pl+i)^2$. Then $\chi(\alpha^{(a)}) = {\rm in}_\omega \bigl( \varphi(\alpha^{(a)})\bigr)$, the initial form of $\varphi(\alpha^{(a)})$, and we have $\,{\rm in}_\prec \bigl( \chi(\alpha^{(a)}) \bigr) \,= \, {\rm in}_\prec \bigl( \varphi(\alpha^{(a)}) \bigr) \,$ for all $\,\alpha^{(a)} \in {\mathcal C}^{np}_{p,m}$. Thus image($\varphi$) and image($\chi$) have the same initial algebra, and so we deduce the sagbi property for the polynomials $\chi(\alpha^{(a)})$ from Theorem \[issagbi\]. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ Let ${\bf w}$ denote the weight on the variables ${\mathcal C}^{np}_{p,m}$ defined by ${\bf w}(\alpha^{(a)}):=-a^2$. For every incomparable pair $\gamma^{(c)},\delta^{(d)}$ in the poset ${\mathcal C}^{np}_{p,m}$, we define the quadratic polynomial $$R(\gamma^{(c)},\delta^{(d)})\quad:=\quad {\rm in}_{\bf w} \bigl( S(\gamma^{(c)},\delta^{(d)}) \bigr),$$ where $S(\gamma^{(c)},\delta^{(d)})$ is the element of the reduced Gröbner basis for the kernel of $\varphi$. For example, $\, R(156^{(1)},234^{(2)})\,$ equals the sum of the first ten terms in (\[BigSyzygy\]). The weight ${\bf w}$ is equivalent, modulo the homogeneities of $\,{\rm kernel}(\varphi)$, to the induced weight which was denoted by ${\mathcal A}^T \omega$ in [@Sturmfels_GBCP Chapter 11]. The only-if direction in [@Sturmfels_GBCP Theorem 11.4] implies \[row-c-GB\] The reduced Gröbner basis of the kernel of $\chi$ consists of the quadratic polynomials $R(\gamma^{(c)},\delta^{(d)})$ as $\gamma^{(c)},\delta^{(d)}$ run over the set of incomparable pairs in the poset ${\mathcal C}^{np}_{p,m}$. For the Plücker ideal defining the classical Grassmannian $(n=0)$, an explicit (but non-reduced) quadratic Gröbner basis is known. It appears in the work of Hodge-Pedoe [@Hodge_Pedoe] and Doubilet-Rota-Stein [@DRS], and it consists of the van der Waerden syzygies. They are discussed in Gröbner basis language in [@Sturmfels_invariant Section 3.1]. Our next aim is to introduce an analogous non-reduced Gröbner basis for the ideal $\, {\rm kernel}(\varphi)\,$ of the quantum Grassmannian. We begin by defining the skew van der Waerden syzygies for its initial ideal $$\label{bfw} \, {\rm kernel}(\chi) \quad = \quad {\rm in}_{\bf w} ({\rm kernel}(\varphi)) ,$$ which consists of the algebraic relations among the row-consecutive minors. Given a sequence of integers $\,D\,: \, 1 \leq d_1 < \cdots < d_p \leq m+p$ and any integer $0\leq a\leq np$, let $D^{(a)}$ denote $\pm \alpha^{(a)}$, where $\alpha$ is the reordering of the sequence $D$ and $\pm$ is the sign of the permutation which sorts the sequence $D$. Let $T=\alpha^{(a)}\beta^{(b)}$ with $a<b$ be a non-standard tableau and $i$ the smallest index of a violation $\beta_i<\alpha_{i-b+a}$. Define increasing sequences $$\begin{array}{c} A\quad:=\quad \alpha_1,\ldots,\alpha_{i-b+a-1} \qquad B\quad:=\quad \beta_{i+1},\ldots,\beta_p\\ C\quad:=\quad \beta_1,\ldots,\beta_i,\alpha_{i-b+a},\ldots,\alpha_p. \end{array}$$ For a subset $I\in\binom{[p+b-a+1]}{i}$, let $C_I$ be the corresponding numbers from $C$ (in order) and $C_{I^c}$ be the other numbers from $C$, also in order. Define the [*skew van der Waerden syzygy*]{} $$\label{eq:ElQuSy} W(T) \quad:=\quad \sum_{I\in\binom{[p+b-a+1]}{i}} (A, C_{I^c})^{(a)}\ \cdot (C_I,B)^{(b)}\ .$$ \[NonRedGB\] The syzygies $\, W(T) \,$ form a Gröbner basis for the kernel of $\chi$. [**Proof.** ]{} Our choice of term order implies $\,{\rm in}_\prec \bigl( W(T) \bigr) = T = \alpha^{(a)} \beta^{(b)}$. Therefore it suffices to show that $\chi(W(T))=0$. Let $Y_1,\ldots,Y_{m+p}$ be the columns of the submatrix of ${\mathcal L}$ given by its rows $a+1,\ldots,b+p$. The skew van der Waerden syzygy $\chi(W(T))$ is an anti-symmetric, multilinear form in the $p+b-a+1$ vectors $Y_{b_1},\ldots,Y_{b_{p+b-a+1}}$ in $(p+b-a)$-space. (7.5,7.5)(0,0) ( 0, 0)[(1,0)[7.5]{}]{} ( 0,2.5)[(1,0)[7.5]{}]{} ( 0, 5)[(1,0)[7.5]{}]{} (2.5,7.5)[(1,0)[5]{}]{} ( 0, 0)[(0,1)[5]{}]{} (2.5, 0)[(0,1)[7.5]{}]{} ( 5, 0)[(0,1)[7.5]{}]{} (7.5, 0)[(0,1)[7.5]{}]{} ( 5, 0) ------------------------------------------------------------------------ The non-reduced Gröbner basis in Proposition \[NonRedGB\] can be lifted to the quantum Grassmannian as follows. We define the [*quantum van der Waerden syzygy*]{} of the non-standard tableau $T$ to be the unique quadratic polynomial $V(T)$ in ${\rm kernel}(\varphi)$ which satisfies $${\rm in}_{\bf w} \bigl( V(T) \bigr ) \quad = \quad W(T),$$ and is a sum of syzygies $S(\gamma^{(c)},\delta^{(d)})$ with ${\bf w}(\gamma^{(c)}\delta^{(d)})={\bf w}(T)$. This syzygy exists by (\[bfw\]) and it is unique because the quadratic generators of the initial ideal are $k$-linearly independent, and any two such quadratic lifts of $W(T)$ in kernel$(\varphi)$ differ by terms whose weights are strictly less than ${\bf w}(T)$. For instance, the quantum van der Waerden syzygy $\, V(156^{(1)}234^{(2)})\,$ is the polynomial with $30$ terms given in (\[BigSyzygy\]). It would be desirable to find an explicit formula, perhaps in terms of the combinatorial formalism in [@DRS], for all of the skew van der Waerden syzygies $V(T)$, but at present we have no clue how to do this. .1cm The ideal of the quantum Grassmannian $K^q_{p,m}$ contains certain [*obvious relations*]{} which are derived from the Grassmannian ${\it Grass}_pk^{m+p}$. For each $\alpha\in\binom{[m+p]}{p}$ consider the polynomial $$g_\alpha(t) \quad = \quad \alpha^{(q)}\cdot t^q+ \cdots + \alpha^{(1)}\cdot t + \alpha^{(0)}\, .$$ Given any quadratic form $F(\alpha)$ in the Plücker ideal defining ${\it Grass}_pk^{m+p}$ and any $0\leq r \leq 2q$, let $F_r$ be the coefficient of $t^r$ in polynomial $F(g_\alpha(t))$. Since $F(g_\alpha(t))$ is a polynomial in $t$ which vanishes identically on $K^q_{p,m}$, each of its coefficients $F_r$ must also vanish on $K^q_{p,m}$. We call the collection of quadratic polynomials $F_r$ as $F$ ranges over a generating set for the Plücker ideal of ${\it Grass}_pk^{m+p}$ the [*obvious relations*]{}. Rosenthal [@Rosen94] showed the following. The obvious relations define $K^q_{p,m}$ set-theoretically, provided $k$ is infinite. When $p,m \leq 2$, the obvious relations coincide with the reduced Gröbner basis of Theorem \[thm:gbasis\], in particular, they generate the ideal of the quantum Grassmannian $K^q_{2,m}$. This is no longer true for $m=p=3$. There are 35 incomparable pairs in ${\mathcal C}^0_{3,3}$, and hence $35$ linearly independent quadrics in the Plücker ideal of ${\it Grass}_3k^6$. These give rise to $35(2q+1)$ linearly independent obvious relations but when $q>0$ there are $35(2q+1)+2q-1$ incomparable pairs in ${\mathcal C}^q_{3,3}$. Thus the obvious relations do not generate the homogeneous ideal of $K^q_{3,3}$. When $q=1$ or $q=2$ then the obvious relations generate the homogeneous ideal of $K^q_{3,3}$ together with an embedded component supported on the irrelevant ideal. Thus the obvious relations define $K^q_{3,3}$ scheme-theoretically, but not ideal-theoretically. It remains an open problem whether the the obvious relations define $K^q_{p,m}$ scheme-theoretically. Batyrev et.al. [@Batyrev] applied the familiar sagbi property for the Grassmannian in the construction of certain pairs of mirror 3-folds from Calabi-Yau complete intersections in Grassmannians. We are optimistic that the results in this paper will be similarly useful for researchers in the fascinating interplay of algebraic geometry and theoretical physics. The classical straightening law for the Grassmannian and its Schubert varieties were the starting point for the general [*standard monomial theory*]{} for flag varieties. For details and references we refer to the recent work on sagbi bases by Gonciulea and Lakshmibai [@GL]. Our results suggest that standard monomial theory might be extended to certain spaces of rational curves in flag varieties generalizing the quantum Grassmannian $K^q_{p,m}$. [99]{} , [ *Conifold transitions and mirror symmetry for [C]{}alabi-[Y]{}au complete intersections in [G]{}rassmannians*]{}, Nuclear Physics B, 514 (1998), pp. 640–666. , [*Quantum [S]{}chubert calculus*]{}, Adv. 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Research at MSRI supported in part by NSF grant DMS-9701755

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