Canonical bases for the quantum group of type $A_r$ and piecewise-linear combinatorics
Canonical bases for the quantum group of type $A_r$ and piecewise-linear combinatorics
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$A_r$ 类型量子群的规范基和分段线性组合
DOI：
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
A. Zelevinsky
中科院分区：
文献类型：
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作者：
A. Berenstein;A. Zelevinsky
这项工作的动机是以下两个问题，从经典的表示理论。(Both问题对于任意的复半单李代数都是有意义的，但由于我们将只处理Ar的情况，所以我们在这个一般性中将它们公式化）。1.在每个不可约的有限维slr+1-模Vλ中构造一个“好”基，它“具体化”了Littlewood-Richardson规则。[3]中给出了这个问题的精确公式;我们将在稍后更详细地解释它。2.在GLr+1的每个多项式表示中构造一个基，使得Weyl群Sr+1的最大元素w 0（被认为是GLr+1的一个元素）通过一个置换（直到一个符号）作用在这个基上，并显式地计算这个置换。这个问题的动机是最近的工作由约翰Stembridge [10]，并提请我们注意他的谈话在耶路撒冷组合数学会议，1993年5月。
This work was motivated by the following two problems from the classical representation theory. (Both problems make sense for an arbitrary complex semisimple Lie algebra but since we shall deal only with the Ar case, we formulate them in this generality). 1. Construct a “good” basis in every irreducible finite-dimensional slr+1-module Vλ, which “materializes” the Littlewood-Richardson rule. A precise formulation of this problem was given in [3]; we shall explain it in more detail a bit later. 2. Construct a basis in every polynomial representation of GLr+1, such that the maximal element w0 of the Weyl group Sr+1 (considered as an element of GLr+1) acts on this basis by a permutation (up to a sign), and explicitly compute this permutation. This problem is motivated by recent work by John Stembridge [10] and was brought to our attention by his talk at the Jerusalem Combinatorics Conference, May 1993.