Specht filtrations and tensor spaces for the Brauer algebra.

Hu, Jun

Journal of Algebraic Combinatorics, Tome 28 (2008) no. 2, pp. 281-312.

Résumé

Summary: Let $m, n\in \Bbb N$. In this paper we study the right permutation action of the symmetric group $\mathfrak S _{2 n}$ mathfrakS_2n on the set of all the Brauer $n$-diagrams. A new basis for the free $\Bbb $Z-module $\mathfrak B _{ n} \mathfrak $B_n spanned by these Brauer $n$-diagrams is constructed, which yields Specht filtrations for $\mathfrak B _{ n} \mathfrak $B_n . For any $2 m$-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}( -2 m) \mathfrak $B_n(-2m) . In particular, we show that it is a $\mathfrak S _{2 n}$ mathfrakS_2n -submodule of $\mathfrak B _{ n}( -2 m) \mathfrak $B_n(-2m) .

Détail

Keywords: keywords Brauer algebra, symmetric group, tensor space