Specht filtrations and tensor spaces for the Brauer algebra

Abstract

Let $m,n\in \mathbb{N}$. In this paper we study the right permutation action of the symmetric group ${\mathfrak{S}}_{2n}$ on the set of all the Brauer $n$-diagrams. A new basis for the free $\mathbb{Z}$-module ${\mathfrak{B}}_n$ spanned by these Brauer $n$-diagrams is constructed, which yields Specht filtrations for ${\mathfrak{B}}_n$. For any $2m$-dimensional vector space $V$ over a field of arbitrary characteristic, we give an explicit and characteristic free description of the annihilator of the $n$-tensor space $V^{\otimes n}$ in the Brauer algebra ${\mathfrak{B}}_n(-2m)$. In particular, we show that it is a ${\mathfrak{S}}_{2n}$-submodule of ${\mathfrak{B}}_n(-2m)$.