Rational tableaux and the tensor algebra of gln
Rational tableaux and the tensor algebra of gln
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有理表格和 gln 的张量代数
DOI：
10.1016/0097-3165(87)90077-x
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发表时间：
1987
期刊：
影响因子：
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通讯作者：
J. Stembridge
中科院分区：
文献类型：
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作者：
J. Stembridge
作为研究GLn(C)的有限维表示的自然组合工具，给出了通常的列严格表的推广(等价于R.C.King的构造)。这些对象被称为有理表象，因为它们对GL的有理表示起着相同的作用，而普通表象对多项式表象则起着同样的作用。给出了有理表的Schensted插入算法的推广，并将其应用于有理表的计数。GLn张量代数中不可约GLn-模的重数还考虑了当第k个张量幂⊗k分解成同时关于GLn和对称群Skn不可约的模时的计数问题。证明了描述这种分解的插入算法的存在性。使用边界条表的推广，其中允许添加和删除边界条，以描述与该分解相关联的字符。对于拉尔根，这些广义的边界条形表有一个简单的结构，允许导出由于Hanlon和Stanley涉及gln⊗k的(拉尔根)分解的恒等式。
A generalization of the usual column-strict tableaux (equivalent to a construction of R. C. King) is presented as a natural combinatorial tool for the study of finite dimensional representations ofGLn(C). These objects are called rational tableaux since they play the same role for rational representations ofGLnas ordinary tableaux do for polynomial representations. A generalization of Schensted's insertion algorithm is given for rational tableaux, and is used to count the. multiplicities of the irreducibleGLn-modules in the tensor algebra ofGLn. The problem of counting multiplicities when thekth tensor powergln⊗kis decomposed into modules which are simultaneously irreducible with respect toGLnand the symmetric groupSkis also considered. The existence of an insertion algorithm which describes this decomposition is proved. A generalization of border strip tableaux, in which both addition and deletion of border strips is allowed, is used to describe the characters associated with this decomposition. For largen, these generalized border strip tableaux have a simple structure which allows derivation of identities due to Hanlon and Stanley involving the (largen) decomposition ofgln⊗k.