On (λ,μ)-tableaux, subsequences and certain elements which determine various cells in Weyl groups of type A

A Multigraph Approach for Performing the Quantum Schur Transform

We take inspiration from the Okounkov-Vershik approach to the representation theory of the symmetric groups to develop a new way of understanding how the Schur-Weyl duality can be used to perform the Quantum Schur Transform. The Quantum Schur Transform is a unitary change of basis transformation between the computational basis of $(\mathbb{C}^d)^{\otimes n}$ and the Schur-Weyl basis of $(\mathbb{C}^d)^{\otimes n}$. We describe a new multigraph, which we call the Schur-Weyl-Young graph, that represents both standard Weyl tableaux and standard Young tableaux in the same diagram. We suggest a maj…

Young tableaux and representations of Hecke algebras of type ADE

We introduce and study some affine Hecke algebras of type ADE, generalising the affine Hecke algebras of GL. We construct irreducible calibrated representations and describe the calibrated spectrum. This is done in terms of new families of combinatorial objects equipped with actions of the corresponding Weyl groups. These objects are built from and generalise the usual standard Young tableaux, and are controlled by the considered affine Hecke algebras. By restriction and limiting procedure, we obtain several combinatorial models for representations of finite Hecke algebras and Weyl groups of t…

经典 Weyl 群的赝次数

Fake degrees of classical Weyl groups

本文借助 domino tableaux 给出经典 Weyl 群各表示 fake degrees 的计算方法。

Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux

This paper makes precise the close connection between the affine Hecke algebra, the path model, and the theory of crystals. Section 2 is a basic pictorial exposition of Weyl groups and affine Weyl groups and Section 5 is an exposition of the theory of (a) symmetric functions, (b) crystals and (c) the path model. Sections 3 and 4 give an exposition of the affine Hecke algebra and recent results regarding the combinatorics of spherical functions on p-adic groups (Hall-Littlewood polynomials). The $q$-analogue of the theory of crystals developed in Section 4 specializes to the path model version …

Cylindrical Combinatorics and Representations of Cherednik Algebras of type A

We investigate the representation theory of the rational and trigonometric Cherednik algebra of type $GL_n$ by means of combinatorics on periodic (or cylindrical) skew diagrams. We introduce and study standard tableaux and plane partitions on periodic diagrams, and in particular, compute some generating functions concerning plane partitions, where Kostka polynomials and their level restricted generalization appear. On representation side, we study representations of Cherednik algebras which admit weight decomposition with respect to a certain commutative subalgebra. All the irreducible represe…

Tableau models for semi-infinite Bruhat order and level-zero representations of quantum affine algebras

We prove that semi-infinite Bruhat order on an affine Weyl group is completely determined from those on the quotients by affine Weyl subgroups associated with various maximal (standard) parabolic subgroups of finite type. Furthermore, for an affine Weyl group of classical type, we give a complete classification of all cover relations of semi-infinite Bruhat order (or equivalently, all edges of the quantum Bruhat graphs) on the quotients in terms of tableaux. Combining these we obtain a tableau criterion for semi-infinite Bruhat order on an affine Weyl group of classical type. As an application…

Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theory

We show that the Young tableaux theory and constructions of the irreducible representations of the Weyl groups of type A, B and D, Iwahori-Hecke algebras of types A, B, and D, the complex reflection groups G(r,p,n) and the corresponding cyclotomic Hecke algebras H_{r,p,n}, can be obtained, in all cases, from the affine Hecke algebra of type A. The Young tableaux theory was extended to affine Hecke algebras (of general Lie type) in recent work of A. Ram. We also show how (in general Lie type) the representations of general affine Hecke algebras can be constructed from the representations of sim…

On Category $\mathcal{O}$ over triangular Generalized Weyl Algebras

We analyze the BGG Category $\mathcal{O}$ over a large class of generalized Weyl algebras (henceforth termed GWAs). Given such a "triangular" GWA for which Category $\mathcal{O}$ decomposes into a direct sum of subcategories, we study in detail the homological properties of blocks with finitely many simples. As consequences, we show that the endomorphism algebra of a projective generator of such a block is quasi-hereditary, finite-dimensional, and graded Koszul. We also classify all tilting modules in the block, as well as all submodules of all projective and tilting modules. Finally, we prese…

On the Combinatorics of Crystal Graphs, I. Lusztig's Involution

In this paper, we continue the development of a new combinatorial model for the irreducible characters of a complex semisimple Lie group. This model, which will be referred to as the alcove path model, can be viewed as a discrete counterpart to the Littelmann path model. It leads to an extensive generalization of the combinatorics of irreducible characters from Lie type A (where the combinatorics is based on Young tableaux, for instance) to arbitrary type. The main results of this paper are: (1) a combinatorial description of the crystal graphs corresponding to the irreducible representations …

A crystal embedding into Lusztig data of type $A$

Let $i$ be a reduced expression of the longest element in the Weyl group of type $A$, which is adapted to a Dynkin quiver with a single sink. We present a simple description of the crystal embedding of Young tableaux of arbitrary shape into $i$-Lusztig data, which also gives an algorithm for the transition matrix between Lusztig data associated to reduced expressions adapted to quivers with a single sink.

某些 PBW 基上晶体结构的组合描述（扩展摘要）

Combinatorial descriptions of the crystal structure on certain PBW bases (extended abstract)

本文研究单复李代数 PBW 基上的无穷晶体，通过限定 Weyl 群最长元素约化表达式的条件，使 ADE 型晶体算子可用简单括号规则描述，并建立其与 A、D 型表格组合学的联系。

A型中的Weyl广群、Young图与Borel子代数

The Weyl groupoid in Type A, Young diagrams and Borel subalgebras

本文在 $m>n$、互素且 $\kappa=-n/m$ 时，用周长 $n$、长度 $m$ 的旋转圆柱上的 Young 图统一刻画 Weyl 广群轨道与仿射 Lie 超代数 Borel 子代数轨道的图同构。

包含光滑元素的Kazhdan-Lusztig右胞腔的一个刻画

A characterization of Kazhdan-Lusztig right cells containing smooth elements

本文刻画了$\mathfrak{sl}(n,\mathbb{C})$的Weyl群$S_n$中包含光滑元素的Kazhdan-Lusztig右胞腔，给出了仅含光滑元素的胞腔的完整分类、仅含非光滑元素及含光滑元素的充分条件，并提出了找出给定胞腔中所有光滑元素的高效算法。