Tableaux on periodic skew diagrams and irreducible representationsof the double affine Hecke algebra of type A
Tableaux on periodic skew diagrams and irreducible representationsof the double affine Hecke algebra of type A
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A型双仿射赫克代数的周期斜图和不可约表示的表格
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发表时间：
2004
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通讯作者：
M. Vazirani
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作者：
Takeshi Suzuki;M. Vazirani
对称群S_n的不可约表示由称为杨图或形状的组合对象来参数化。一个给定的不可约表示有一个由该形状的Young表索引的基。实际上，这个基由群代数FSN的交换子代数F[X]的权向量(同时特征向量)组成。双仿射Hecke代数(DAHA)是仿射对称群的群代数的变形，它还包含一个交换子代数F[X]。并非每个Daha的不可约表示都有权向量的基(事实上，将其所有的不可约表示都参数化是相当困难的)，但如果我们将注意力限制在那些有权向量的基础上，这些不可约表示就被“仿射形状”参数化，并且具有由该形状的“仿射表格”索引的(X-权向量的)基。在这次演讲中，我们将构建这些不可约表示。导言。我们引入并研究了斜杨图的仿射模拟和图上的画面。A型双仿射Hecke代数作用于每个图上的标准画面所跨越的空间。我们证明了用这种方法得到的模是不可约的，并且它们耗尽了双仿射Hecke代数上某一类的所有不可约模。特别地，恢复了Cherednik所宣布的这类不可约模的分类。众所周知，由n个框组成的Young图将对称群S_n的有限维不可约表示的同构类参数化，而且每个不可约表示的结构都用对应的Young图上的Tableaux来描述，即表示的一个基用标准Tableaux来标记，并在其上显式地描述了S_n生成元的作用。A.Young的这种组合刻划在对称群(或仿射Hecke代数)的表示理论的研究中起到了至关重要的作用，并在[Ch1，Ra1，Ra2]中给出了它对Gln的(退化)仿射Hecke代数Hn(Q)的推广，其中斜Young图出现在组合侧。给出域F上Gln的双仿射Hecke代数Ḧn(Q)的一族不可约表示的参数表示和组合描述，其中Q∈F是该代数的一个参数.双仿射Hecke代数是由I.Cherednik[CH2，CH3]引入的，后来被他和几位作者用来得到关于对角不变量、Macdonald多项式和某些Macdonald恒等式的重要结果。在本文中，我们主要讨论q不是1的根的情形，并且我们考虑X-半单的Ḧn(Q)的表示，即我们考虑
The irreducible representations of the symmetric group Sn are parameterized by combinatorial objects called Young diagrams, or shapes. A given irreducible representation has a basis indexed by Young tableaux of that shape. In fact, this basis consists of weight vectors (simultaneous eigenvectors) for a commutative subalgebra F[X] of the group algebra FSn. The double affine Hecke algebra (DAHA) is a deformation of the group algebra of the affine symmetric group and it also contains a commutative subalgebra F[X]. Not every irreducible representation of the DAHA has a basis of weight vectors (and in fact it is quite difficult to parameterize all of its irreducible representations), but if we restrict our attention to those that do, these irreducible representations are parameterized by “affine shapes” and have a basis (of X-weight vectors) indexed by the “affine tableaux” of that shape. In this talk, we will construct these irreducible representations. Introduction. We introduce and study an affine analogue of skew Young diagrams and tableaux on them. The double affine Hecke algebra of type A acts on the space spanned by standard tableaux on each diagram. We show that the modules obtained this way are irreducible, and they exhaust all irreducible modules of a certain class over the double affine Hecke algebra. In particular, the classification of irreducible modules of this class, announced by Cherednik, is recovered. As is well-known, Young diagrams consisting of n boxes parameterize isomorphism classes of finite dimensional irreducible representations of the symmetric group Sn, and moreover the structure of each irreducible representation is described in terms of tableaux on the corresponding Young diagram; namely, a basis of the representation is labeled by standard tableaux, on which the action of Sn generators is explicitly described. This combinatorial description due to A. Young has played an essential role in the study of the representation theory of the symmetric group (or the affine Hecke algebra), and its generalization for the (degenerate) affine Hecke algebra Hn(q) of GLn has been given in [Ch1, Ra1, Ra2], where skew Young diagrams appear on combinatorial side. The purpose of this paper is to introduce an “affine analogue” of skew Young diagrams and tableaux, which give a parameterization and a combinatorial description of a family of irreducible representations of the double affine Hecke algebra Ḧn(q) of GLn over a field F, where q ∈ F is a parameter of the algebra. The double affine Hecke algebra was introduced by I. Cherednik [Ch2, Ch3] and has since been used by him and by several authors to obtain important results about diagonal coinvariants, Macdonald polynomials, and certain Macdonald identities. In this paper, we focus on the case where q is not a root of 1, and we consider representations of Ḧn(q) that are X-semisimple; namely, we consider representations which