A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs
A combinatorial realization of Schur-Weyl duality via crystal graphs and dual equivalence graphs
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通过晶体图和对偶等价图组合实现 Schur-Weyl 对偶性
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发表时间：
2008
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通讯作者：
Sami H. Assaf
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作者：
Sami H. Assaf
对于特殊线性群的任何多项式表示，对应晶体的节点可以通过半标准杨氏表索引。在某些条件下，出现标准的Young tableaux，并且权重为0。标准Young tableaux也参数化对偶等价图的顶点。受表象理论的启发，本文通过Schur-Weyl对偶的组合形式来解释这种联系。特别是，我们把一个对偶等价图结构的0-重量空间的某些晶体图，生产边组合从晶体边缘。这种构造可以用Stembridge对晶体图和作者对对偶等价图给出的局部刻画来表示。
For any polynomial representation of the special linear group, the nodes of the corresponding crystal may be indexed by semi-standard Young tableaux. Under certain conditions, the standard Young tableaux occur, and do so with weight 0. Standard Young tableaux also parametrize the vertices of dual equivalence graphs. Motivated by the underlying representation theory, in this paper, we explainthis connection by giving a combinatorial manifestation of Schur-Weyl duality. In particular, we put a dual equivalence graph structure on the 0-weight space of certain crystal graphs, producing edges combinatorially from the crystal edges. The construction can be expressed in terms of the local characterizations given by Stembridge for crystal graphs and the author for dual equivalence graphs.