On a theorem of Lehrer and Zhang.

zbMATH Open1281.20051arXiv1105.5287MaRDI QIDQ455488

Jun Hu, Zhan Kui Xiao

Publication date: 22 October 2012

Published in: Documenta Mathematica (Search for Journal in Brave)

Abstract: Let $K$ be an arbitrary field of characteristic not equal to 2. Let $m,ninN$ and $V$ an $m$ dimensional orthogonal space over $K$. There is a right action of the Brauer algebra $$ on the $n$-tensor space $V^{otimesn}$ which centralizes the left action of the orthogonal group $O(V)$. Recently G.I. Lehrer and R.B. Zhang defined certain quasi-idempotents $E_i$ in $$ (see ( ef{keydfn})) and proved that the annihilator of $V^{otimesn}$ in $$ is always equal to the two-sided ideal generated by $E_{[(m+1)/2]}$ if $chK=0$ or $chK>2(m+1)$. In this paper we extend this theorem to arbitrary field $K$ with $chKeq2$ as conjectured by Lehrer and Zhang. As a byproduct, we discover a combinatorial identity which relates to the dimensions of Specht modules over symmetric groups of different sizes and a new integral basis for the annihilator of $V^{otimesm+1}$ in $$.

Full work available at URL: https://arxiv.org/abs/1105.5287

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zbMATH Keywords

idempotentssymmetric groupsorthogonal groupsirreducible representationsmultiplicitiesBrauer algebrasSchur-Weyl dualitystandard tableauxtensor spacesdimensions of Specht modules

Mathematics Subject Classification ID

Combinatorial aspects of representation theory (05E10) Hecke algebras and their representations (20C08) Representations of finite symmetric groups (20C30) Representation theory for linear algebraic groups (20G05) Vector and tensor algebra, theory of invariants (15A72) Representations of quivers and partially ordered sets (16G20)

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