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

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 $\mathfrak{g}\mathfrak{l}(m\mathrm{\mid }n)$. 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.