RoCK blocks for double covers of symmetric groups and quiver Hecke superalgebras

Abstract

Chapter 1. Introduction ix 1.1. Broué's conjecture for symmetric groups and beyond ix 1.2. RoCK blocks for double covers of symmetric groups xii 1.3. Quiver Hecke superalgebras xvii Chapter 2. Background material 2.1. Basic notation 2.2. Graded superalgebras and supermodules 2.3. Combinatorics 2.4. Lie theory Chapter 3. Quiver Hecke superalgebras 3.1. Quiver Hecke superalgebras and a dimension formula 3.2. Further properties of quiver Hecke superalgebras 3.3. Cuspidality Chapter 4. RoCK blocks of quiver Hecke superalgebras 4.1. RoCK blocks 4.2. Gelfand-Graev truncation of Rδ and Z ρ,1 4.3. Gelfand-Graev truncation of Rdδ 4.4. Intertwiners 4.5. Affine zigzag relations Chapter 5. RoCK blocks of double covers of symmetric groups 5.1. Twisted wreath superproducts 5.2. Blocks of double covers 5.3. Kang-Kashiwara-Tsuchioka isomorphism 5.4. Broué's conjecture for RoCK blocks of double covers Chapter 6. Appendix. Some calculations in B 2 6.1. Some small rank computations 6.2. A commutation lemma 6.

Derived superequivalences for spin symmetric groups and odd sl(2)-categorifications

“…C n where C n is the Clifford superalgebra. Building off of foundational work of Kleshchev and Livesey [33], and categorical actions introduced in [23,24], our derived equivalences for blocks of cyclotomic quiver Hecke superalgebras for Λ = Λ 0 implies the abelian defect conjecture for spin representations of the symmetric group (Theorem 6.3). 1.4.…”

“…The superblocks B (ρ,d) of kS − n are labelled by pairs (ρ, d), where ρ is a p-core partition (see section [33,Section 2.3] and [34]) and d is a non-negative integer such that |ρ| + dp = n. It is known that the defect group D (ρ,d) of B (ρ,d) is abelian when d < p. Let b (ρ,d) be the Brauer correspondent of the block D (ρ,d) . Assume then that d < p, then Broué's defect conjecture for the spin symmetric group states that the block B (ρ,d) with abelian defect group D (ρ,d) is derived equivalent to its Brauer correspondent b (ρ,d) .…”

“…As explained in [33], unlike the blocks k S n (1 − e z ) ∼ = kS n that are relatively well understood, very little is known about spin blocks of symmetric groups, and in particular, Broué's conjecture for them is wide open. In this article, we prove the spin defect conjecture by utilizing so-called odd categorical sl 2 -actions.…”

“…The authors are grateful to Raphael Rouquier for helpful conversations. They are also grateful to Jon Brundan and Sasha Kleshchev for discussing their work in progress and clarifying aspects of Kleshchev-Livesey [33]. The second author first became aware of the connection between the spin symmetric group and odd categorifications through conversations with Jon and Sasha.…”

“…When Λ = Λ 0 , The Kang, Kashiwara, Oh categorification theorem [23] implies that R Λ0 β is nonzero, if and only if Λ 0 − β is a nonzero weight of V (Λ 0 ). Nonzero weights of V (Λ 0 ) can be understood in terms of the set of p-strict partitions λ utilizing its residue content cont(λ) ∈ Q + , see [33,Section 2.1d]. Expressed in terms of the p-core ρ of the p-strict partition λ, the nonzero weights of V (Λ 0 ) are of the form cont(ρ) + dδ for some p-core partition ρ and d ∈ ≥0 .…”