Dual partially harmonic tensors and quantized Schur--Weyl duality

TL;DR

This paper establishes a surjective homomorphism linking certain quotients of the Birman--Murakami--Wenzl algebra to endomorphism algebras of tensor powers of symplectic spaces, using diagram categories and canonical bases.

Contribution

It proves the surjectivity of a natural homomorphism between algebraic quotients and endomorphism algebras in the context of quantized Schur--Weyl duality for symplectic spaces.

Findings

Surjective homomorphism from algebra quotient to endomorphism algebra

Use of diagram category of framed tangles and canonical basis

Extension of Schur--Weyl duality to partially harmonic tensors

Abstract

Let $V$ be a $2m$-dimensional symplectic space over an infinite field $K$. Let $\mathfrak{B}^{(f)}_{n,K}$ be the two-sided ideal of the Birman--Murakami--Wenzl algebra $\mathfrak{B}_{n,K}$ generated by $E_1E_3\cdots E_{2f-1}$ with $1\leq f\leq\left\lfloor \frac n2 \right\rfloor$. In this paper, using the diagram category of framed tangles and canonical basis, we prove that the natural homomorphism from $\mathfrak{B}_{n,K}/\mathfrak{B}^{(f)}_{n,K}$ to $ \mathrm{End}_{U_q(\mathfrak{sp}_{2m})}\left(V^{\otimes n}/\left(V^{\otimes n}\cdot \mathfrak{B}^{(f)}_{n,K}\right)\right)$ is always surjective.