The odd Littlewood-Richardson rule

A bijection between Littlewood-Richardson tableaux and rigged configurations

A bijection is defined from Littlewood-Richardson tableaux to rigged configurations. It is shown that this map preserves the appropriate statistics, thereby proving a quasi-particle expression for the generalized Kostka polynomials, which are q-analogues of multiplicities in tensor products of irreducible general linear group modules indexed by rectangular partitions.

Littlewood-Richardson coefficients for reflection groups

In this paper we explicitly compute all Littlewood-Richardson coefficients for semisimple or Kac-Moody groups G, that is, the structure coefficients of the cohomology algebra H^*(G/P), where P is a parabolic subgroup of G. These coefficients are of importance in enumerative geometry, algebraic combinatorics and representation theory. Our formula for the Littlewood-Richardson coefficients is given in terms of the Cartan matrix and the Weyl group of G. However, if some off-diagonal entries of the Cartan matrix are 0 or -1, the formula may contain negative summands. On the other hand, if the Cart…

标志Littlewood-Richardson表与经典群的分支规则

Flagged Littlewood-Richardson tableaux and branching rule for classical groups

本文推广Littlewood限制公式，提出了从${\ m GL}_n$到${\ m O}_n$分支规则的新公式，该公式基于带标志条件的Littlewood-Richardson表，并据此给出了$B_n$和$D_n$型Lusztig $t$-权重重数的组合公式。

Jack Littlewood-Richardson 系数与 Nazarov-Sklyanin Lax 算子

Jack Littlewood-Richardson Coefficients and the Nazarov-Sklyanin Lax Operator

本文研究 Nazarov-Sklyanin 量子 Lax 算子多项式本征函数性质，通过考虑本征函数乘积推广 Kerov 公式，推导出基于分区乘法运算的 Jack Littlewood-Richardson 系数约束系统。

约化Schur函数与Littlewood-Richardson系数

Reduced Schur Functions and the Littlewood-Richardson Coefficients

研究了约化Schur函数。阐明了它们与$A^(1)_{r-1}$的基本表示和对称群的模表示的关系。Littlewood-Richardson系数出现在约化Schur函数之间的线性关系中。

From Littlewood-Richardson coefficients to cluster algebras in three lectures

This is an expanded version of the notes of my three lectures at a NATO Advanced Study Institute ``Symmetric functions 2001: surveys of developments and perspectives" (Isaac Newton Institute for Mathematical Sciences, Cambridge, UK; June 25-July 6, 2001). Lecture I presents a unified expression due to A. Berenstein and the author for generalized Littlewood-Richardson coefficients (= tensor product multiplicities) for any complex semisimple Lie algebra. Lecture II outlines a proof of this result; the main idea of the proof is to relate the LR-coefficients with canonical bases and total positivi…

基本A_1^（1）-模的权向量与Littlewood-Richardson规则

Weight Vectors of the Basic A_1^(1)-Module and the Littlewood-Richardson Rule

研究了$\A$的基本表示。权向量用Schur函数表示。给出了任意权空间的一个合适的基。Littlewood-Richardson规则出现在权向量之间的线性关系中。

Crystal duality and Littlewood-Richardson rule of extremal weight crystals

We consider a category of $\gl_\infty$-crystals, whose objects are disjoint unions of extremal weight crystals of non-negative level with certain finite conditions on the multiplicity of connected components. We show that it is a monoidal category under tensor product of crystals and the associated Grothendieck ring is anti-isomorphic to an Ore extension of the character ring of integrable lowest weight $\gl_\infty$-modules with respect to derivations shifting the characters of fundamental modules. A Littlewood-Richardson rule of extremal weight crystals with non-negative level is described ex…

A cyclage poset structure for Littlewood-Richardson tableaux

A graded poset structure is defined for the sets of Littlewood-Richardson (LR) tableaux that count the multiplicity of an irreducible GL(n)-module in the tensor product of irreducibles indexed by a sequence of rectangular partitions. This poset generalizes the cyclage poset on column-strict tableaux defined by Lascoux and Schutzenberger, and its grading function generalizes the charge statistic. It is shown that the polynomials obtained by enumerating LR tableaux by shape and the generalized charge, are the Poincare polynomials of isotypic components of the certain modules supported in the clo…

A Littlewood-Richardson Rule for factorial Schur functions

We give a combinatorial rule for calculating the coefficients in the expansion of a product of two factorial Schur functions. It is a special case of a more general rule which also gives the coefficients in the expansion of a skew factorial Schur function. Multiplication rules for the Capelli operators and quantum immanants are also given.

Littlewood-Richardson Coefficients via Yang-Baxter Equation

The purpose of this paper is to present an interpretation for the decomposition of the tensor product of two or more irreducible representations of GL(N) in terms of a system of quantum particles. Our approach is based on a certain scattering matrix that satisfies a Yang-Baxter type equation. The corresponding piecewise-linear transformations of parameters give a solution to the tetrahedron equation. These transformation maps are naturally related to the dual canonical bases for modules over the quantum enveloping algebra $U_q(sl_n)$. A byproduct of our construction is an explicit description …

斜Schur模的多参数变形的根为1的Littlewood-Richardson滤子。

A Littlewood-Richardson filtration at roots of 1 for multiparameter deformations of skew Schur modules.

设R是交换环，q是R的单位，P是系数为q的整数次幂的乘反对称矩阵。用SE（q，P）表示与q和P有关的多参数量子矩阵双代数.对[Hashimoto-Hayashi，Tohoku Math.Tohoku Math.J. 44（1992）]作了一点推广，定义了经典斜Schur模的多参数变形$L_{ł/μ}V_P$.当R是域且q不是1的根时，像[H-H]中给出的参数表明$L_{ł/μ}V_P$是不可约的，其分解为不可约的是$\sum_νc(ł/μ;ν)L_νV_P$，其中系数是通常的Littlewood-Richardson系数。当R是任意环且q是1的根时，我们构造了$L_{ł/μ}V_P$的一个滤子作为SE（q，P）-余模，使得它的分次对象恰好是$\sum_νc(ł/μ;ν)L_νV_P$ .

Demazure Characters and Affine Fusion Rules

The Demazure character formula is applied to the Verlinde formula for affine fusion rules. We follow Littelmann's derivation of a generalized Littlewood-Richardson rule from Demazure characters. A combinatorial rule for affine fusions does not result, however. Only a modified version of the Littlewood-Richardson rule is obtained that computes an (old) upper bound on the fusion coefficients of affine $A_r$ algebras. We argue that this is because the characters of simple Lie algebras appear in this treatment, instead of the corresponding affine characters. The Bruhat order on the affine Weyl gro…