A 2-basic set of the alternating group

TL;DR

This paper constructs a 2-basic set for the alternating group \\A_n by extending a 2-basic set from the symmetric group \\sym_n, utilizing generalized perfect isometries and adapting existing methods for odd characteristic.

Contribution

It introduces a new construction of a 2-basic set for \\A_n by leveraging a specialized 2-basic set of \\sym_n and applying generalized perfect isometries.

Findings

Successfully constructed a 2-basic set for \\A_n.

Extended methods from symmetric to alternating groups.

Utilized generalized perfect isometries in the construction.

Abstract

In this note, we construct a 2-basic set of the alternating group \A_n. To do this, we construct a 2-basic set of the symmetric group \sym_n with an additional property, such that its restriction to \A_n is a 2-basic set. We adapt here a method developed in \cite{BrGr} for the case when the characteristic is odd. One of the main tools is the generalized perfect isometries defined by K\"ulshammer, Olsson and Robinson in \cite{KOR}.