Dual partially harmonic tensors and quantized Schur--Weyl duality

Abstract

Let $V$ be a $2m$-dimensional symplectic space over an infinite field $K$. Let ${\mathfrak{B}}_{n,K}^{(f)}$ be the two-sided ideal of the Birman--Murakami--Wenzl algebra ${\mathfrak{B}}_{n,K}$ generated by $E_1{E}_3\cdots E_{2f-1}$ with $1\le f\le \lfloor \frac{n}{2}\rfloor$. In this paper, using the diagram category of framed tangles and canonical basis, we prove that the natural homomorphism from ${\mathfrak{B}}_{n,K}/{\mathfrak{B}}_{n,K}^{(f)}$ to ${\mathrm{End}}_{U_q({\mathfrak{sp}}_{2m})}\left(V^{\otimes n}/\left(V^{\otimes n}\cdot {\mathfrak{B}}_{n,K}^{(f)}\right)\right)$ is always surjective.

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