On a theorem of Lehrer and Zhang

TL;DR

This paper extends Lehrer and Zhang's theorem on the annihilator of tensor spaces in Brauer algebras to all fields with characteristic not equal to 2, revealing new combinatorial identities and bases.

Contribution

It generalizes a key theorem about Brauer algebra actions on tensor spaces to arbitrary fields with characteristic not 2, confirming a conjecture by Lehrer and Zhang.

Findings

Extended Lehrer and Zhang's theorem to all fields with characteristic ≠ 2.

Discovered a new combinatorial identity related to Specht modules.

Introduced a new integral basis for the annihilator in the Brauer algebra.

Abstract

Let $K$ be an arbitrary field of characteristic not equal to 2. Let $m, n\in\N$ and $V$ an $m$ dimensional orthogonal space over $K$. There is a right action of the Brauer algebra $\bb_n(m)$ on the $n$-tensor space $V^{\otimes n}$ 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 $\bb_n(m)$ (see (\ref{keydfn})) and proved that the annihilator of $V^{\otimes n}$ in $\bb_n(m)$ is always equal to the two-sided ideal generated by $E_{[(m+1)/2]}$ if $\ch K=0$ or $\ch K>2(m+1)$. In this paper we extend this theorem to arbitrary field $K$ with $\ch K\neq 2$ 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 $V^{\otimes m+1}$ in $\bb_{m+1}(m)$.