Bipartite extension graphs and the Duflo--Serganova functor

Abstract

We consider several examples when the extension graph admits a bipartition compatible with the action of Duflo-Serganova functors.

Gruson-Serganova character formulas and the Duflo-Serganova cohomology functor

“…given by ds x : sch N → sch DS x (N); moreover, ds x coincides with the evaluation of sch to a subalgebra h x ⊂ h. In Theorem 7.2 we show that ds x (E − L ) is given by a simple formula (for the exceptional Lie superalgebras a similar formula follows from [Gor2]). Since DS x preserves sdim, this gives a formula for sdim E − L , see Corollary 7.2.8.…”

“…Since the graph is a Z-graded, these matrices are lower-triangular with a > λ,λ = a < λ,λ = 1. The following lemma is proven in [Gor2], Section 3.4 (the proof is similar to one in [GS1], Thm. 4).…”

“…Using (3) and the forementioned formulas one obtains the expressions for sch DS x (L), sdim L, [L : L ′ ] and dim L for L ∈ Irr (we do not write this long expressions). Note that DS x (L) is described in [HW], [GH] and [Gor2]; various formulas for dim L and sdim L appeared in [SuZh1], [M], [HW],and [GH].…”

“…The proof of ( 12) follows the plan explained in [Gor2], Section 3. In 5.1, 5.2 below we recall the main constructions of [Gor2].…”

“…We take g := gl(m|n) and fix a central character χ. We define Γχ st and its subgraph Γ χ st similarly to [Gor2].…”