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

Abstract

In this paper, we study the representations of the periplectic Lie superalgebra using the Duflo-Serganova functor. Given a simple p(n)-module L and a certain element x ∈ p(n) of rank 1, we give an explicit description of the composition factors of the p(n − 1)-module DS x (L), which is defined as the homology of the complexIn particular, we show that this 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 p(n)-module, based on its highest weight.In particular, this reproves the Kac-Wakimoto conjecture for p(n), which was proved earlier by the authors in [ES19].

The Duflo-Serganova functor, vingt ans après

“…For p(n) these functors are known not to be semisimple, while for q(n) this remains an open question. However by the work of [ES2], the composition factors of DS x (L) for a simple module L are known in some cases for p(n). The case of q(n) will be considered in forthcoming work.…”

“…For the classical series gl(m|n), osp(m|2n), and p(n), there is a remarkable thread which links the computations of the composition factors of DS x L for a simple module L, namely arc diagrams. (Note that for p(n) the diagrams used in [ES2] were called cap diagrams;…”

“…Example 12.26 (The following example is taken from [ES2]). Let n = 9, and consider the dominant weight λ = ǫ 3 + 3ǫ 4 + 3ǫ 5 + 6ǫ 6 + 8ǫ 7 + 8ǫ 8 + 8ǫ 9 .…”

“…The DS functor has a large and growing list of applications throughout the literature. It was used in [S2] to prove the Kac-Wakimoto conjecture (see Section 7); in [IRS] to describe the supercharacter ring for p(n) (see Section 8); in [HPS] to study important sl(∞)-modules (see Section 9); in [ES2] to give a formula for the superdimension of p(n)-modules; in [HsW] to give a new proof of the superdimension formula for GL(m|n)-modules; in [Hs] to obtain reductive envelopes of certain supergroups; in [BKN2] to compute complexity of certain modules over gl(m|n); in [CH] to classify the indecomposable summands of tensor powers of the standard representation of OSP (m|2n); and in [EHS] to construct universal tensor categories. The DS functor has been applied to study Deligne categories in numerous papers (see e.g., [CH,EhSt2,EHS,ES1]).…”

Parabolic category $\mathcal O^{\mathfrak p}$ for periplectic Lie superalgebras $\mathfrak{pe}(n)$

“…While the category F n and the BGG category O for pe(n) have been extensively studied in recent years (see, e.g., [B+9], [B+92], [BK], [CC], [Ch], [Co], [ES1], [ES2], [ES3], [Go], [HIR], [IRS], [Mo], [Se2]), however, the study about parabolic version of BGG category for pe(n) remains to be unavailable in the literature.…”