Dual partially harmonic tensors and quantized Schur--Weyl duality

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 ${\mathfrak{B}}_{n,K}\mathrm{/}{\mathfrak{B}}_{n,K}^{(f)}$ to ${\mathrm{E}\mathrm{n}\mathrm{d}}_{U_q({\mathfrak{s}\mathfrak{p}}_{2m})}\left(V^{\otimes n}\mathrm{/}(V^{\otimes n}\cdot {\mathfrak{B}}_{n,K}^{(f)})\right)$ is always surjective.