We study Hilbert-Samuel multiplicity for points of Schubert varieties in the complete flag variety, by Groebner degenerations of the Kazhdan-Lusztig ideal. In the covexillary case, we give a positive combinatorial rule for multiplicity by establishing (with a Groebner basis) a reduced and equidimensional limit whose Stanley-Reisner simplicial complex is homeomorphic to a shellable ball or sphere. We show that multiplicity counts the number of facets of this complex. We also obtain a formula for the Hilbert series of the local ring. In particular, our work gives a multiplicity rule for Grassmannian Schubert varieties, providing alternative statements and proofs to formulae of [Lakshmibai-Weyman '90], [Rosenthal-Zelevinsky '01], [Krattenthaler '01], [Kreiman-Lakshmibai '04] and [Woo-Yong '09]. We suggest extensions of our methodology to the general case.
V. Lakshmibai and J. Weyman @cite_37 and V. Kreiman and V. Lakshmibai @cite_15 utilized standard monomial theory to determine multiplicity rules for Grassmannians (actually, @cite_37 deduces a recursive rule valid for any minuscule @math ).
@cite_37 @cite_15
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Abstract Elementary proofs are given for theorems of Bapat and Raghavan on the scaling of nonnegative multidimensional matrices. Theorems of Sinkhorn and of Brualdi, Parter, and Schneider are derived as corollaries. For positive two-dimensional matrices, Hilbert's projective metric and a theorem of G. Birkhoff are used to prove that Sinkhorn's original iterative procedure converges geometrically; the ratio of convergence is estimated from the given data. Whereas formal category theory is classically considered within a @math -category, in this paper a double-dimensional approach is taken. More precisely we develop such theory within the setting of hypervirtual double categories, a notion extending that of virtual double category by adding cells with nullary target. [...] After this the notion of weak' Kan extension within a hypervirtual double category is considered, together with three strengthenings. [...] The notion of yoneda embedding is then considered in a hypervirtual double category, and compared to that of a good yoneda structure on a @math -category; the latter in the sense of Street-Walters and Weber. Conditions are given ensuring that a yoneda embedding @math defines @math as the free small cocompletion of @math , in a suitable sense. In the second half we consider formal category theory in the presence of algebraic structures. In detail: to a monad @math on a hypervirtual double category @math several hypervirtual double categories @math of @math -algebras are associated, [...]. This is followed by the study of the creation of, amongst others, left Kan extensions by the forgetful functors @math . The main motivation of this paper is the description of conditions ensuring that yoneda embeddings in @math lift along these forgetful functors, as well as ensuring that such lifted algebraic yoneda embeddings again define free small cocompletions, now in @math . As a first example we apply the previous to monoidal structures on categories, hence recovering Day convolution of presheaves and Im-Kelly's result on free monoidal cocompletion, as well as obtaining a "monoidal Yoneda lemma". Kleene's theorem on the coincidence of regular and rational languages in free monoids has been generalized by Schutzenberger to a description of the recognizable formal power series in noncommuting variables over arbitrary semirings and by Ochmanski to a characterization of the recognizable languages in trace monoids. We will describe the recognizable formal power series over arbitrary semirings and in partially commuting variables, i.e. over trace monoids. We prove that the recognizable series are certain rational power series, which can be constructed from the polynomials by using the operations sum, product, and a restricted star which is applied only to series for which the elements in the support all have the same connected alphabet. The converse is true if the underlying semiring is commutative. Moreover, if in addition the semiring is idempotent then the same result holds with a star restricted to series for which the elements in the support have connected (possibly different) alphabets. It is shown that these assumptions over the semiring are necessary. This provides a joint generalization of Kleene's, Schutzenberger's and Ochmanski's theorems. We show a directed and robust analogue of a boolean isoperimetric-type theorem of Talagrand [Geom. Funct. Anal., 3 (1993), pp. 295--314]. As an application, we give a monotonicity testing algorithm that makes @math nonadaptive queries to a function @math , always accepts a monotone function, and rejects a function that is @math -far from being monotone with constant probability.