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

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 certain translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for $\mathfrak{p}(n)$ resembling those for $\mathfrak{gl}(m \vert n)$. We discover two natural highest weight structures. Using the Temperley–Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of irreducibles in standard and costandard modules and classify the blocks of $\mathcal{F}_n$. We also prove the surprising fact that indecomposable projective modules in this category are multiplicity-free.