Dual partially harmonic tensors and quantized Schur--Weyl duality
Pei Wang, Zhankui Xiao
TL;DR
This work constructs and analyzes a quantized Schur--Weyl duality for dual partially harmonic tensors in $V^{\otimes n}$, with $V$ a $2m$-dimensional symplectic space, by using the diagrammatic framework of framed tangles and the type $C$ RT-functor. It identifies the BMW ideal filtration $V^{\otimes n}\mathfrak{B}^{(f)}_{n,K}$ with truncations $\mathcal{O}_{\pi_f}(V^{\otimes n})$, and shows each layer $V^{\otimes n}\mathfrak{B}^{(f)}_{n,K}/V^{\otimes n}\mathfrak{B}^{(f+1)}_{n,K}$ is isomorphic to the dual of the quantum harmonic tensor $\mathcal{HT}_{f,q}^{\otimes n}$, which carries a Weyl filtration. The central result is the surjectivity of the map $\varphi_f: (\mathfrak{B}_{n,K}/\mathfrak{B}^{(f)}_{n,K})^{\rm op}\to \mathrm{End}_{U_q(\mathfrak{sp}_{2m})}(V^{\otimes n}/V^{\otimes n}\mathfrak{B}^{(f)}_{n,K})$, establishing a quantized type $C$ Schur--Weyl duality for these partially harmonic quotients. The paper also develops base-change invariance and filtrations (good/Weyl) for these objects, ensuring the endomorphism dimensions are independent of the base field and the quantum parameter $q$. Overall, it extends the classical Brauer--Schur--Weyl framework to a robust quantized setting for dual partially harmonic tensors via diagrammatic and categorical techniques.
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.