Duflo-Serganova functor and superdimension formula for the periplectic Lie superalgebra

TL;DR

This paper investigates the representations of the periplectic Lie superalgebra using the Duflo-Serganova functor, providing explicit descriptions of module composition factors and a combinatorial superdimension formula, confirming the Kac-Wakimoto conjecture.

Contribution

It offers a detailed analysis of the Duflo-Serganova functor for rak{p}(n), including explicit composition factor descriptions and a superdimension formula, advancing understanding of superalgebra representations.

Findings

Modules are multiplicity-free after applying the functor.

Derived a combinatorial superdimension formula based on highest weight.

Reproved the Kac-Wakimoto conjecture for rak{p}(n).

Abstract

In this paper, we study the representations of the periplectic Lie superalgebra using the Duflo-Serganova functor. Given a simple $\mathfrak{p}(n)$-module $L$ and a certain element $x\in \mathfrak{p}(n)$ of rank $1$, we give an explicit description of the composition factors of the $\mathfrak{p}(n-1)$-module $DS_x(L)$, which is defined as the homology of the complex $$\Pi M \xrightarrow{x} M \xrightarrow{x} \Pi M.$$ In particular, we show that this $\mathfrak{p}(n-1)$-module is multiplicity-free. We then use this result to give a simple explicit combinatorial formula for the superdimension of a simple integrable finite-dimensional $\mathfrak{p}(n)$-module, based on its highest weight. In particular, this reproves the Kac-Wakimoto conjecture for $\mathfrak{p}(n)$, which was proved earlier by the authors.