Perverse equivalence in SL(2,q)

Abstract

We show that the derived equivalence between full defect blocks of $SL(2,q)$ in defining characteristic and their Brauer correspondants can be presented as a composition of perverse equivalences.

Introduction

Broué's Abelian Defect Conjecture [2, 6.2, Question] is a very important focal point of the block theory of finite groups:

Conjecture

Let $\mathbb{F}$ be an algebraically closed field of characteristic $p>0$. Let G be a finite group and A be a block of $\mathbb{F}\mathit{G}$. If A has an abelian defect group D, then A is derived equivalent to a block B of $N_G(D)$, its Brauer correspondent.

For general introduction, see [11]. This conjecture has been studied by many and it is proved when D is cyclic [15]. However, for the general abelian case it is still a case-by-case verification. For $SL(2,q)$ in defining characteristic, the principal block of Broué's conjecture was proven by Okuyama [13] using a construction that differs from most of the other cases. Yoshii generalised Okuyama's method to the non-principal block case in [20]. As time passed, tools such as mutation or perverse equivalence came into play. In particular, perverse equivalence gathered some geometrical information about certain derived equivalences in surprising ways. One such example is Craven's application of perverse equivalence with Lusztig's L-function in [5]. This paper report a similar job for $SL(2,q)$, by showing the derived equivalence between full defect blocks of $SL(2,q)$ and their Brauer correspondents contemplated by Okuyama (and Yoshii) is a composition of perverse equivalences.

Theorem

(Theorem 5.1) The derived equivalence between full defect blocks of $SL(2,q)$ in defining characteristic and its Brauer correspondent, as introduced by [13] and [20], is a composition of perverse equivalences.

Conventions

We use right modules throughout the paper. Let $\mathbb{F}$ be an algebraically closed field of characteristic p. When A is an finite dimensional (fd) $\mathbb{F}$-algebra, let $A\mathrm{-}\mathrm{mod}$ be the category of finitely-generated (fg) A-modules, $\mathrm{st}(A)$ be the stable category of fg A-modules and $D^b(A)$ be the bounded derived category of fg A-modules. The symbol Ω is used to denote the Heller translate of a module. For complexes we use the cochain convention, that is, differential maps in a complex are of degree 1. A tensor product without a subscript is over the base field $\mathbb{F}$. An asterisk as a superscript over a module denotes its $\mathbb{F}$-dual.

Acknowledgement. This project grew out from the lecture by Radha Kessar during the workshop Morita Equivalence Problems for Blocks of Finite Groups at École Polytechnic Fédérale de Lausanne, September 2016. The author is thus deeply grateful for the organisers and Bernoulli Centre in EPFL. In addition, the author would like to thank Joseph Chuang for the immensely valuable discussions, comments and encouragement during and after his supervision. The author was supported by a JSPS International Research Fellowship (P17814) and is grateful to Aaron Chan and Osamu Iyama's interest and comments for the presentations related to this project during the fellowship.