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First layer formulas for characters of $ {\rm SL}(n,{\bf C})$

Author: John R. Stembridge
Journal: Trans. Amer. Math. Soc. 299 (1987), 319-350
MSC: Primary 20G05; Secondary 05A15, 17B10, 22E46
MathSciNet review: 869415
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Abstract: Some problems concerning the decomposition of certain characters of $ SL(n,\,{\mathbf{C}})$ are studied from a combinatorial point of view. The specific characters considered include those of the exterior and symmetric algebras of the adjoint representation and the Euler characteristic of Hanlon's so-called ``Macdonald complex.'' A general recursion is given for computing the irreducible decomposition of these characters. The recursion is explicitly solved for the first layer representations, which are the irreducible representations corresponding to partitions of $ n$. In the case of the exterior algebra, this settles a conjecture of Gupta and Hanlon. A further application of the recursion is used to give a family of formal Laurent series identities that generalize the (equal parameter) $ q$-Dyson Theorem.

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Keywords: Symmetric functions, Schur functions, group characters, weight vectors, exterior algebra, symmetric algebra, general linear group, special linear group, $ q$-Dyson Theorem
Article copyright: © Copyright 1987 American Mathematical Society

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