A 2-basic set of the alternating group

Abstract

In this note, we construct a 2-basic set of the alternating group \({\mathfrak{A}_n}\). To do this, we construct a 2-basic set of the symmetric group \({\mathfrak{S}_n}\) with an additional property, such that its restriction to \({\mathfrak{A}_n}\) is a 2-basic set. We adapt here a method developed by Brunat and Gramain (J. Reine Angew. Math., to appear) for the case when the characteristic is odd. One of the main tools is the generalized perfect isometries defined by Kulshammer et al. (Invent. Math. 151, 513–552, (2003)).