作为Littlewood-Richardson表亲对偶版本的基座表（socle tableau）

Generalised Stretched Littlewood-Richardson Coefficients

The Littlewood-Richardson (LR) coefficient counts among many other things the LR tableaux of a given shape and a given content. We prove, that the number of LR tableaux weakly increases if one adds to the shape and the content the shape and the content of another LR tableau. We also investigate the behaviour of the number of LR tableaux, if one repeatedly adds to the shape another shape with either fixed or arbitrary content. This is a generalisation of the stretched LR coefficients, where one repeatedly adds the same shape and content to itself.

On the largest Kronecker and Littlewood--Richardson coefficients

We give new bounds and asymptotic estimates for Kronecker and Littlewood--Richardson coefficients. Notably, we resolve Stanley's questions on the shape of partitions attaining the largest Kronecker and Littlewood--Richardson coefficients. We apply the results to asymptotics of the number of standard Young tableaux of skew shapes.

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。

Box moves on Littlewood-Richardson tableaux and an application to invariant subspace varieties

In his 1951 book "Infinite Abelian Groups", Kaplansky gives a combinatorial characterization of the isomorphism types of embeddings of a cyclic subgroup in a finite abelian group. In this paper we first use partial maps on Littlewood-Richardson tableaux to generalize this result to finite direct sums of such embeddings. We then focus on an application to invariant subspaces of nilpotent linear operators. We develop a criterion to decide if two irreducible components in the representation space are in the boundary partial order.

The entries in the LR-tableau

Let $Γ$ be the Littlewood-Richardson tableau corresponding to an embedding $M$ of a subgroup in a finite abelian $p$-group. Each individual entry in $Γ$ yields information about the homomorphisms from $M$ into a particular subgroup embedding, and hence determines the position of $M$ within the category of subgroup embeddings. Conversely, this category provides a categorification for LR-tableaux in the sense that all subgroup embeddings corresponding to a given LR-tableau share certain homological properties.

Littlewood-Richardson系数与Kazhdan-Lusztig多项式

Littlewood-Richardson coefficients and Kazhdan-Lusztig polynomials

证明了仿射对称群的Littlewood-Richardson系数是某些抛物型Kazhdan-Lusztig多项式在1处的值.这些q-类似物的Littlewood-Richardson多重性与以前介绍的丝带tableaux。

一个超 Littlewood--Richardson 型规则

A super Littlewood--Richardson type rule

本文提出了带符号字母表上超 Schur 函数的超 Littlewood--Richardson 规则，并利用超 Young 表的性质给出了相应系数的组合解释，为表示论等领域提供了应用工具。

The odd Littlewood-Richardson rule

In previous work with Mikhail Khovanov and Aaron Lauda we introduced two odd analogues of the Schur functions: one via the combinatorics of Young tableaux (odd Kostka numbers) and one via the odd symmetrization operator. In this paper we introduce a third analogue, the plactic Schur functions. We show they coincide with both previously defined types of Schur function, confirming a conjecture. Using the plactic definition, we establish an odd Littlewood-Richardson rule. We also re-cast this rule in the language of polytopes, via the Knutson-Tao hive model.

Vanishing of Littlewood-Richardson polynomials is in P

J. DeLoera-T. McAllister and K. D. Mulmuley-H. Narayanan-M. Sohoni independently proved that determining the vanishing of Littlewood-Richardson coefficients has strongly polynomial time computational complexity. Viewing these as Schubert calculus numbers, we prove the generalization to the Littlewood-Richardson polynomials that control equivariant cohomology of Grassmannians. We construct a polytope using the edge-labeled tableau rule of H. Thomas-A. Yong. Our proof then combines a saturation theorem of D. Anderson-E. Richmond-A. Yong, a reading order independence property, and E. Tardos' algo…

The co-Pieri rule for Kronecker coefficients

A fundamental problem in the representation theory of the symmetric group, Sn, is to describe the coefficients in the decomposition of a tensor product of two simple representations. These coefficients are known in the literature as the Kronecker coefficients. The Littlewood--Richardson coefficients appear as an important subfamily of the wider class of stable Kronecker coefficients. This subfamily of coefficients can be calculated using a tableaux counting algorithm known as the Littlewood--Richardson rule. This paper generalises one half of this rule (the "co-Pieri" rule) to the the wider fa…

几何Littlewood-Richardson规则

A geometric Littlewood-Richardson rule

我们描述了一个明确的几何Littlewood-Richardson规则，解释为变形的交叉两个舒伯特品种，使他们打破舒伯特品种。对基场没有任何限制，所有出现的多重数都是1;这对应用程序很重要。这条规则应该被看作是皮耶里规则的推广到任意舒伯特类，通过显式同伦。它与其他Littlewood-Richardson规则有一个直接的双射，例如tableaux，Knutson和Tao的谜题。这给出了Littlewood-Richardson规则的第一个几何证明和解释。它有一个主机的几何后果，在配套文件“舒伯特归纳”。这个规则在K理论中也有一个解释，由Buch提出，它给了K理论一个扩展的谜题。该规则建议一个自然的方法来解决一个悬而未决的问题，找到一个Littlewood-Richardson规则的旗帜品种，导致一个猜想，证明是真实的5维。最后，该规则为类似的开放问题提供了方法，例如辛格拉斯曼和双旗簇的Littlewood-Richardson规则。

Linear time equivalence of Littlewood--Richardson coefficient symmetry maps

Benkart, Sottile, and Stroomer have completely characterized by Knuth and dual Knuth equivalence a bijective proof of the conjugation symmetry of the Littlewood-Richardson coefficients. Tableau-switching provides an algorithm to produce such a bijective proof. Fulton has shown that the White and the Hanlon-Sundaram maps are versions of that bijection. In this paper one exhibits explicitly the Yamanouchi word produced by that conjugation symmetry map which on its turn leads to a new and very natural version of the same map already considered independently. A consequence of this latter construct…

GL_n张量积代数的一个基

A Basis for the GL_n Tensor Product Algebra

本文主要研究$GL_n$张量积代数，它封装了$GL_n$的任意有限维不可约表示的张量积的分解.我们将描述这个代数的一个显式基。这种结构直接关系到Littlewood-Richardson系数的Littlewood-Richardson表的组合描述。在哲学上，人们可以把这种构造看作是在经典不变理论的背景下对利特尔伍德-理查森规则的重铸。

社会层最高权模与经典类型Richardson轨道的刻画

A characterization of socular highest weight modules and Richardson orbits of classical types

本文刻画经典类型单复李代数中社会层最高权模，通过Z-图描述相应Richardson轨道的划分类型。

辛Kashiwara-Nakashima表aux的余晶体、辛Willis直接法、虚拟键及其应用

Cocrystals of symplectic Kashiwara-Nakashima tableaux, symplectic Willis like direct way, virtual keys and applications

本文提出了一种计算辛KN表aux左右键的余晶体方法，结合Baker虚拟化嵌入技术，并应用于测试旗簇上辛KN表aux的标准性。