Jun Hu, Zhankui Xiao

摘要

Let $K$ be an arbitrary field of characteristic not equal to $2$. Let $m, ninN$ 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 $V^{otimes n}$ 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 $b_n(m)$ (see (ef{keydfn})) and proved that the annihilator of $V^{otimes n}$ in $b_n(m)$ is always equal to the two-sided ideal generated by $E_{[(m+1)/2]}$ if $ch K=0$ or $ch K2(m+1)$. In this paper we extend this theorem to arbitrary field $K$ with $ch Keq 2$ 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 $b_{m+1}(m)$.