We study a class of systems of functional equations closely related to various kinds of integrable statistical and quantum mechanical models. We call them the finite and infinite @math -systems according to the number of functions and equations. The finite Q-systems appear as the thermal equilibrium conditions (the Sutherland–Wu equation) for certain statistical mechanical systems. Some infinite Q-systems appear as the relations of the normalized characters of the KR modules of the Yangians and the quantum affine algebras. We give two types of power series formulae for the unique solution (resp. the unique canonical solution) for a finite (resp. infinite) Q-system. As an application, we reformulate the Kirillov–Reshetikhin conjecture on the multiplicities formula of the KR modules in terms of the canonical solutions of Q-systems.
11 @cite_2 . Chari proved the type III formula of @math for @math for any @math for the classical algebras, and for some @math for the exceptional algebras.
@cite_2
1554718120 2023541349 2069546133 1835694908
We give an algebro-combinatorial proof of a general ver­ sion of Pieri's formula following the approach developed by Fomin and Kirillov in the paper "Quadratic algebras, Dunkl elements, and Schu­ bert calculus." We prove several conjectures posed in their paper. As a consequence, a new proof of classical Pieri's formula for cohomol­ ogy of complex flag manifolds, and that of its analogue for quantum cohomology is obtained in this paper. An arithmetic formula is multilinear if the polynomial computed by each of its subformulas is multilinear. We prove that any multilinear arithmetic formula for the permanent or the determinant of an n × n matrix is of size super-polynomial in n. Previously, super-polynomial lower bounds were not known (for any explicit function) even for the special case of multilinear formulas of constant depth. We consider arithmetic formulas consisting of alternating layers of addition (+) and multiplication (×) gates such that the fanin of all the gates in any fixed layer is the same. Such a formula Φ which additionally has the property that its formal syntactic degree is at most twice the (total) degree of its output polynomial, we refer to as a regular formula. As usual, we allow arbitrary constants from the underlying field F on the incoming edges to a + gate so that a + gate can in fact compute an arbitrary F-linear combination of its inputs. We show that there is an (n2 + 1)-variate polynomial of degree 2n in VNP such that any regular formula computing it must be of size at least nΩ(log n). Along the way, we examine depth four (ΣΠΣΠ) regular formulas wherein all multiplication gates in the layer adjacent to the inputs have fanin a and all multiplication gates in the layer adjacent to the output node have fanin b. We refer to such formulas as ΣΠ[b]ΣΠ[a]-formulas. We show that there exists an n2-variate polynomial of degree n in VNP such that any ΣΠ[O(√n)]ΣΠ[√n]-formula computing it must have top fan-in at least 2Ω(√n·log n). In comparison, Tavenas [Tav13] has recently shown that every nO(1)-variate polynomial of degree n in VP admits a ΣΠ[O(√n)]ΣΠ[√n]-formula of top fan-in 2O(√n·log n). This means that any further asymptotic improvement in our lower bound for such formulas (to say 2ω(√n log n)) will imply that VP is different from VNP. Let @math be a primitive Hilbert modular form of parallel weight @math and level @math for the totally real field @math , and let @math be a rational prime coprime to @math . If @math is ordinary at @math and @math is a CM extension of @math of relative discriminant @math prime to @math , we give an explicit construction of the @math -adic Rankin-Selberg @math -function @math . When the sign of its functional equation is @math , we show, under the assumption that all primes @math are principal ideals of @math which split in @math , that its central derivative is given by the @math -adic height of a Heegner point on the abelian variety @math associated with @math . This @math -adic Gross--Zagier formula generalises the result obtained by Perrin-Riou when @math and @math satisfies the so-called Heegner condition. We deduce applications to both the @math -adic and the classical Birch and Swinnerton-Dyer conjectures for @math .