First layer formulas for characters of 𝑆𝐿(𝑛,𝐶)

Stembridge, John R.

doi:10.1090/S0002-9947-1987-0869415-X

First layer formulas for characters of $\textrm {SL}(n,\textbf {C})$
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by John R. Stembridge

Trans. Amer. Math. Soc. 299 (1987), 319-350

DOI: https://doi.org/10.1090/S0002-9947-1987-0869415-X

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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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