Dual partially harmonic tensors and Brauer-Schur-Weyl duality

TL;DR

This paper explores the structure of partially harmonic tensors in symplectic tensor spaces, establishing their dimensions, filtrations, and dualities with Brauer algebra ideals, revealing deep connections between symplectic representation theory and algebraic structures.

Contribution

It introduces a new understanding of partially harmonic tensors, their Weyl filtrations, and their relations to Brauer algebra ideals, extending the duality framework in symplectic tensor representations.

Findings

Dimensions of tensor subspaces are independent of the field.

Natural homomorphism from Brauer algebra quotient to endomorphism algebra is surjective.

Constructed filtrations reveal the structure of tensor modules and their dualities.

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.