Translation functors and decomposition numbers for the periplectic Lie superalgebra $\mathfrak{p}(n)$

TL;DR

This paper explores the representation theory of the periplectic Lie superalgebra $rak{p}(n)$, introducing new combinatorial tools and algebraic actions to compute module multiplicities and classify blocks in its category of finite-dimensional representations.

Contribution

It defines an action of the Temperley--Lieb algebra on the category and introduces weight and arrow diagrams for $rak{p}(n)$, enabling explicit calculations of decomposition numbers.

Findings

Calculated multiplicities of standard and costandard modules.

Classified the blocks of the category $F_n$.

Proved indecomposable projective modules are multiplicity-free.

Abstract

We study the category $\mathcal{F}_n$ of finite-dimensional integrable representations of the periplectic Lie superalgebra $\mathfrak{p}(n)$. We define an action of the Temperley--Lieb algebra with infinitely many generators and defining parameter $0$ on the category $\mathcal{F}_n$ by translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for $\mathfrak{p}(n)$ resembling those for $\mathfrak{gl}(m|n)$. Using the Temperley--Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of standard and costandard modules in indecomposable projective modules and classify the blocks of $\mathcal{F}_n$. We also prove that indecomposable projective modules in this category are multiplicity-free.