Jacobi-Trudi type formula for character of irreducible representations of $\frak{gl}(m|1)$

TL;DR

This paper establishes a determinantal formula for calculating irreducible characters of the Lie superalgebra rak{gl}(m|1), generalizing the Jacobi-Trudi formula and confirming a conjecture for rak{gl}(m|n).

Contribution

It provides the first explicit determinantal formula for rak{gl}(m|1) characters, extending classical results to superalgebras and confirming a conjecture for rak{gl}(m|n).

Findings

Derived a determinantal formula for rak{gl}(m|1) characters.

Confirmed the conjecture by van der Jeugt and Moens for rak{gl}(m|n).

Generalized the Jacobi-Trudi formula to Lie superalgebras.

Abstract

We prove a determinantal type formula to compute the irreducible characters of the general Lie superalgebra $\mathfrak{gl}(m|1)$ in terms of the characters of the symmetric powers of the fundamental representation and their duals. This formula was conjectured by J. van der Jeugt and E. Moens for the Lie superalgebra $\frak{gl}(m|n)$ and generalizes the well-known Jacobi-Trudi formula.