Two Boundary Centralizer Algebras for $\mathfrak{gl}(n|m)$

TL;DR

This paper constructs a new algebraic action related to the general linear Lie superalgebra and shows that certain module decompositions remain irreducible under this action, extending understanding of symmetries in superalgebra representations.

Contribution

It introduces an action of the degenerate two boundary braid algebra on modules over (n|m) and demonstrates irreducibility preservation under a specific quotient algebra.

Findings

Action of (n|m) modules extends to a quotient algebra ^{ext}_d.

Irreducible summands remain irreducible under the quotient algebra.

Provides new tools for analyzing symmetries in superalgebra representations.

Abstract

We define an action of the degenerate two boundary braid algebra $\mathcal{G}_d$ on the $\mathbb{C}$-vector space $M\otimes N\otimes V^{\otimes d}$, where $M$ and $N$ are arbitrary modules for the general linear Lie superalgebra $\mathfrak{gl}(n|m)$, and $V$ is the natural representation. When $M$ and $N$ are parametrized by rectangular hook Young diagrams, this action factors through a quotient $\mathcal{H}^{\operatorname{ext}}_d$. The irreducible summands of $M\otimes N\otimes V^{\otimes d}$ for the centralizer of $\mathfrak{gl}(n|m)$, remain irreducible once regarded as modules for this quotient $\mathcal{H}^{\operatorname{ext}}_d$.