Canonical bases for the Brauer centralizer algebra
Canonical bases for the Brauer centralizer algebra
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DOI：
10.4310/mrl.1995.v2.n1.a3
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发表时间：
1995
影响因子：
1
通讯作者：
S. Fishel;I. Grojnowski
中科院分区：
文献类型：
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作者：
S. Fishel;I. Grojnowski
在本文中，我们构建了 Birman-Wenzl 代数 BWn（布劳尔扶正代数的 q 类似物）的规范基，从而定义了左、右和两侧单元。我们组合地描述这些对象（推广对称群的 Robinson-Schensted 算法）并表明每个左侧单元格都带有 BWn 的不可约表示。特别是，我们获得了在 Z 上定义的每个表示的规范基。相同的技术可推广到任意缠结代数和 Rmatrix [R]；特别是量子群作用于 V ⊗r 的中心化器，因为 V 是量子群的有限维表示。 BWn 出现在参数 (q, r, x) 的特定值上，作为 Uqsp2k 或 Uqok 对其标准表示 V 的 n 次张量幂的作用的集中器。人们可能会转移 BWn 模块的基来给出 V ⊗n 中出现的表示的基（如 [GL] 中），并且很自然地推测如此获得的基与 [L,§27] 的基一致。在 Weyl 群中，只有在对称群中，单元表示是不可约的。在这方面，BWn 与 Sn 类似。由于与量子群的关系，人们会预料到这一点，量子群的行为也类似于 A 型赫克代数 [L]。此外，我们对 BWn 结构的主要新见解正是这种形式——我们证明每个表示都是以精确的方式从对称群的表示中导出的（参见§6.5）。除了呼吁解决 [KL,1.4] 中 Sn 的相应问题之外，这篇论文本质上是独立的。特别是，除了将其描述为
In this paper we construct canonical bases for the Birman-Wenzl algebra BWn, the q-analogue of the Brauer centralizer algebra, and so define left, right and two-sided cells. We describe these objects combinatorially (generalizing the Robinson-Schensted algorithm for the symmetric group) and show that each left cell carries an irreducible representation of BWn. In particular, we obtain canonical bases for each representation, defined over Z. The same technique generalizes to an arbitrary tangle algebra and Rmatrix [R]; in particular to centralizers of the quantum group action on V ⊗r, for V a finite dimensional representation of a quantum group. BWn occurs for particular values of the parameters (q, r, x) as the centralizers of the action of Uqsp2k or Uqok on the n-th tensor power of its standard representation V . One may presumably transfer the bases of the BWn modules to give a basis of representations occurring in V ⊗n (as in [GL]), and it is natural to conjecture that the basis so obtained coincides with that of [L,§27]. Of the Weyl groups, only in the symmetric group are the cell representations irreducible. In this respect BWn is similar to Sn. One would expect this because of the relation with quantum groups, which also behave like Hecke algebras of type A [L]. Moreover, our main new insight into the structure of BWn is precisely of this form—we show that every representation is induced from a representation of a symmetric group in a precise way (see §6.5). This paper is essentially self-contained, except for an appeal to the solution of the corresponding problem for Sn in [KL,1.4]. In particular, we make no further mention of quantum groups and use no previous work on the structure of BWn (e.g. [BW,HR,W]) except for its description as a