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

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