Ribbon tiling and character formula for periplectic Lie superalgebras

TL;DR

This paper provides a combinatorial character formula for finite-dimensional irreducible representations of the periplectic Lie superalgebra, using ribbon tilings of skew Young diagrams to express characters as cancellation-free sums.

Contribution

It introduces a novel combinatorial approach to compute characters of periplectic Lie superalgebra modules via ribbon tilings, extending previous methods.

Findings

Character formula expressed as cancellation-free sum over Kac modules

Ribbon tilings correspond to terms in the character formula

Provides explicit combinatorial tools for representation theory of $rak{p}(n)$

Abstract

We give a combinatorial formula for the character of a finite-dimensional irreducible representation of the periplectic Lie superalgebra $\mathfrak{p}(n)$. The character of irreducible module $L(\mu)$ is given by a cancellation-free alternating sum over the characters of thick or thin Kac modules, $\Delta(\lambda)$ or $\nabla(\lambda)$, such that there exists a ribbon tiling of a skew Young diagram $\lambda/\mu$.