Brou\'e's perfect isometry conjecture holds for the double covers of the symmetric and alternating groups

TL;DR

This paper proves Broué's perfect isometry conjecture for blocks of double covers of symmetric and alternating groups with weight less than p, extending previous results and explicitly constructing related characters.

Contribution

It establishes Broué's perfect isometry conjecture for these double covers when the block weight is less than p and constructs the characters of their Brauer correspondents.

Findings

Broué's conjecture holds for double covers with weight less than p

Explicit construction of Brauer correspondent characters

Extension of previous isometries to new cases

Abstract

O. Brunat and J. Gramain recently proved that any two blocks of double covers of symmetric groups are Brou\'{e} perfectly isometric provided they have the same weight and sign. They also proved a corresponding statement for double covers of alternating groups and Brou\'{e} perfect isometries between double covers of symmetric and alternating groups when the blocks have opposite signs. Using both the results and methods of O. Brunat and J. Gramain in this paper we prove that when the weight of a block of a double cover of a symmetric or alternating group is less than $p$ then the block is Brou\'{e} perfectly isometric to its Brauer correspondent. This means that Brou\'{e}'s perfect isometry conjecture holds for the double covers of the symmetric and alternating groups. We also explicitly construct the characters of these Brauer correspondents which may be of independent interest to the reader.