Dual partially harmonic tensors and Brauer-Schur-Weyl duality

Abstract

Let $V$ be a $2m$-dimensional symplectic vector space over an algebraically closed field $K$. Let \mbb_n^{(f)} be the two-sided ideal of the Brauer algebra \mbb_n(-2m) over $K$ generated by $e_1e_3... e_{2f-1}$, where $0\leq f\leq [n/2]$. Let $\mathcal{HT}_{f}^{\otimes n}$ be the subspace of partially harmonic tensors of valence $f$ in $V^{\otimes n}$. In this paper, we prove that $\dim\mathcal{HT}_f^{\otimes n}$ and \dim\End_{KSp(V)}\Bigl(V^{\otimes n}/V^{\otimes n}\mbb_n^{(f)}\Bigr) are both independent of $K$, and the natural homomorphism from \mbb_n(-2m)/\mbb_n^{(f)} to \End_{KSp(V)}\Bigl(V^{\otimes n}/V^{\otimes n}\mbb_n^{(f)}\Bigr) is always surjective. We show that $\mathcal{HT}_{f}^{\otimes n}$ has a Weyl filtration and is isomorphic to the dual of V^{\otimes n}\mbb_n^{(f)}/V^{\otimes n}\mbb_n^{(f+1)} as a $Sp(V)$-(\mbb_n(-2m)/\mbb_n^{(f+1)})-bimodule. We obtain a $Sp(V)$-\mbb_n-bimodules filtration of $V^{\otimes n}$ such that each successive quotient is isomorphic to some \nabla(\lam)\otimes z_{g,\lam}\mbb_n with \lam\vdash n-2g, \ell(\lam)\leq m and $0\leq g\leq [n/2]$, where \nabla(\lam) is the co-Weyl module associated to \lam and z_{g,\lam} is an explicitly constructed maximal vector of weight \lam. As a byproduct, we show that each right \mbb_n-module z_{g,\lam}\mbb_n is integrally defined and stable under base change