Super Vust theorem and Schur-Sergeev duality for principal finite $W$-superalgebras

TL;DR

This paper extends classical theorems to Lie superalgebras, establishing a super Vust theorem and Schur-Sergeev duality for principal finite W-superalgebras, enriching the understanding of superalgebra representations.

Contribution

It formulates a super Vust theorem and derives a Schur-Sergeev duality for principal finite W-superalgebras, advancing superalgebra representation theory.

Findings

Super Vust theorem formulated for $rak{gl}(m|n)$

Schur-Sergeev duality established for principal finite W-superalgebras

Partial super version of Brundan-Kleshchev's higher level Schur-Weyl duality

Abstract

Considering the general linear Lie superalgebra $\mathfrak{gl}(m|n)=\mathfrak{gl}(m|n)_{\bar{\bar 0}}\oplus \mathfrak{gl}(m|n)_{\bar{\bar 1}}$ over $\mathbb{C}$, we first formulate a super version of Vust theorem associated with a principal nilpotent element $e\in \mathfrak{gl}(m|n)_{\bar{\bar 0}}$. As an application of this theorem, we then obtain a Schur-Sergeev duality for principal finite $W$-superalgebras which is partially a super version of Brundan-Kleshchev's higher level Schur-Weyl duality established in \cite{BKl}