TY - UNPB

T1 - Duflo-Serganova functor and superdimension formula for the periplectic Lie superalgebra

AU - Entova-Aizenbud, Inna

AU - Serganova, Vera

PY - 2019

Y1 - 2019

N2 - 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.

AB - 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.

KW - Mathematics - Representation Theory

M3 - פרסום מוקדם

T3 - Arxiv preprint

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

ER -