A 2-basic set of the alternating group

Abstract

In this note, we construct a 2-basic set of the alternating group An. To do this, we construct a 2-basic set of the symmetric group Sn with an additional property, such that its restriction to An 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 Külshammer et al. (Invent. Math. 151, 513-552, (2003)). Mathematics Subject Classification (2000). Primary 20C15; Secondary 20C20, 20C30.

Basic Sets for the Double Covering Groups of the Symmetric and Alternating Groups in Odd Characteristic

“…While it is unclear how to exhibit a p-basic set of G in general, Hiss conjectured that all finite groups do have p-basic sets. This conjecture is true in the following cases: for p-soluble groups [11, X.2.1], for certain finite reductive groups under various assumptions ( [17], [12], [15], [13], [14], [22], [10], [6]), and for symmetric and alternating groups ( [21], [9], [8], [2]).…”

Submatrices of character tables and basic sets

“…In the more recent investigation of generalized blocks for the symmetric groups, also characters on ℓ-regular classes where ℓ is not necessarily prime have been studied; this is closely connected to the theory of Hecke algebras at roots of unity. For some results on basic sets for finite groups of Lie type see [6,10,11,12], for results on symmetric and alternating groups see [2,7,8,15], [14,Sec. 6.3].…”