Lie Superalgebras and Generalized Kazhdan-Lusztig Polynomials

TL;DR

The paper introduces `liesuperalg`, a SageMath package that facilitates advanced calculations in Lie superalgebra representation theory, including characters, invariants, and Kazhdan-Lusztig polynomials, specifically for Type A.

Contribution

It provides new computational tools for Lie superalgebra representations, including algorithms for module decomposition and combinatorics of generalized Kazhdan-Lusztig polynomials.

Findings

Implemented an algorithm for module decomposition into irreducibles.

Enabled calculation of characters and invariants of weights.

Facilitated combinatorial analysis of Kazhdan-Lusztig polynomials.

Abstract

We present `liesuperalg` a SageMath package for representation-theoretic calculations involving Lie superalgebras in Type A. Our package introduces functionality to calculate invariants of weights and produce the associated cup diagrams. We expose functionality to calculate characters of irreducible representations, work with combinatorics of generalized Kazhdan-Lusztig polynomials, and determine composition factor multiplicities of indecomposable modules. Our package implements an algorithm to decompose arbitrary modules in terms of irreducible ones in the Grothendeick group of Lie superalgebra representations.