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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The multiplicity of the Steinberg representation of $ {\rm GL}\sb n{\bf F}\sb q$ in the symmetric algebra


Authors: N. J. Kuhn and S. A. Mitchell
Journal: Proc. Amer. Math. Soc. 96 (1986), 1-6
MSC: Primary 20G40; Secondary 20G05, 20J06, 55R40, 55S10
MathSciNet review: 813797
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Abstract: Let $ S(V)$ denote the symmetric algebra on the standard $ n$-dimensional representation $ V$ of $ {\text{G}}{{\text{L}}_n}{{\mathbf{F}}_q}$. The multiplicity series in $ S(V)$ for the Steinberg representation St of $ {\text{G}}{{\text{L}}_n}{{\mathbf{F}}_q}$ is determined. This series is defined by $ {F_{{\text{S}}\text{t}}}(t) = \sum\nolimits_{k = 0}^\infty {{a_k}{t^k}} $ where $ a_k$ is the multiplicity of St in the $ k$th symmetric power $ {S^k}(V)$. We show that $ {F_{{\text{S}}t}}(t) = {t^r}\prod\nolimits_{i = 1}^n {{{(1 - {t^{{q^i} - 1}})}^{ - 1}}} $, where $ r = \sum\nolimits_{i = 1}^{n - 1} {({q^i} - 1} )$. The proof involves a general property of Tits buildings and a computation of the invariants in $ S(V)$ of the parabolic subgroups of $ {\text{G}}{{\text{L}}_n}{{\mathbf{F}}_q}$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0813797-6
PII: S 0002-9939(1986)0813797-6
Article copyright: © Copyright 1986 American Mathematical Society