sl_2-actions along short strings for spin blocks

TL;DR

This paper proposes a new approach to defining certain operators in the context of spin blocks of symmetric and alternating groups, focusing on short strings and their tilting complexes, with implications for source algebra equivalences.

Contribution

It introduces a novel method for defining E_i and F_i operators along short strings, incorporating paired simples and crossing over between groups, inspired by permutation modules.

Findings

Operators are halved when both halves are isomorphic.

The approach is compatible with existing tilting complexes.

The method applies to individual strings in the spin block context.

Abstract

The problem of source algebra equivalences between blocks at the ends of the maximal strings of spin blocks of the symmetric and alternating groups has recently been settled, but so far there has not even been a candidate of the tilting complex defining the reflections of the internal blocks of the string. Our aim in this paper is to propose a definition for the mappings E_i and F_i and for their divided powers. The solution we propose here is to halve the operators only when both halves are isomorphic, which, on the level of the Grothendieck group, corresponds to working with paired simples as pairs, and crossing over between the symmetric and alternating groups, in order to send paired simples to paired simples.This is equivalent to working with the irreducible supermodules, as in the work of Brundan and Kleshchev, though we don't introduce the supermodule language into this paper. The main difference between our approach and theirs is the crossovers, which occasionally force extra halving of modules.The definition is based on the method of permutation m odules from Kessar-Schaps. We apply it to short strings and show that, at least on the individual strings, it is compatible with the form of the tilting complex used by Chuang and Rouquier.