# On a theorem of Lehrer and Zhang

### Jun Hu

### Zhankui Xiao

## Abstract

Let $K$ be an arbitrary field of characteristic not equal to 2. Let $m,n∈N$ and $V$ be an $m$ dimensional orthogonal space over $K$. There is a right action of the Brauer algebra $B_{n}(m)$ on the $n$-tensor space $V_{⊗n}$ 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 $B_{n}(m)$ (see (1.1)) and proved that the annihilator of $V_{⊗n}$ in $B_{n}(m)$ is always equal to the two-sided ideal generated by $E_{[(m+1)/2]}$ if $charK=0$ or $charK>2(m+1)$. In this paper we extend this theorem to arbitrary field $K$ with $charK=2$ as conjectured by Lehrer and Zhang. As a byproduct, we discover a combinatorial identity which relates to the dimensions of Specht modules over the symmetric groups of different sizes and a new integral basis for the annihilator of $V_{⊗m+1}$ in $B_{m+1}(m)$.

## Cite this article

Jun Hu, Zhankui Xiao, On a theorem of Lehrer and Zhang. Doc. Math. 17 (2012), pp. 245–270

DOI 10.4171/DM/367