Simple character formulas for finite $W$-superalgebras of type $A$

TL;DR

This paper derives a canonical basis character formula for irreducible modules in categories related to finite W-superalgebras of type A, unifying various character formulas in Lie superalgebra and W-algebra theory.

Contribution

It introduces a uniform character formula for modules over finite W-superalgebras of type A, generalizing previous formulas in related algebraic categories.

Findings

Established a canonical basis character formula for irreducible modules.

Categorified tensor product modules of polynomial representations and their duals.

Unified character formulas for Lie superalgebras and W-algebras of type A.

Abstract

We establish a canonical basis character formula for the irreducible modules in arbitrary parabolic BGG-type categories, including the category of finite-dimensional modules, for finite $W$-superalgebras of type $A$. These categories categorify the tensor product modules of irreducible polynomial representations and their duals over a quantum group of type $A$. Moreover, the standard modules and irreducible modules in these categories categorify the standard basis and Lusztig's dual canonical basis in the tensor product modules. Our formula provides a uniform generalization of several character formulas in BGG categories for Lie superalgebras and for $W$-algebras of type $A$.