On a theorem of Lehrer and Zhang

Abstract

Let $K$ be an arbitrary field of characteristic not equal to 2. Let $m,n\in \mathbb{N}$ and $V$ be an $m$ dimensional orthogonal space over $K$. There is a right action of the Brauer algebra ${\mathfrak{B}}_n(m)$ 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 ${\mathfrak{B}}_n(m)$ (see (1.1)) and proved that the annihilator of $V^{\otimes n}$ in ${\mathfrak{B}}_n(m)$ is always equal to the two-sided ideal generated by $E_{[(m+1)/2]}$ if $\mathrm{char}K=0$ or $\mathrm{char}K>2(m+1)$. In this paper we extend this theorem to arbitrary field $K$ with $\mathrm{char}K\ne 2$ 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 $V^{\otimes m+1}$ in ${\mathfrak{B}}_{m+1}(m)$.