A basic set for the alternating group

TL;DR

This paper proves the existence of p-basic sets for the alternating group _n for odd primes p, using generalized perfect isometries, and explores implications for decomposition numbers in representation theory.

Contribution

It establishes the existence of p-basic sets for _n for odd primes p, extending known results from symmetric groups using generalized perfect isometries.

Findings

Existence of p-basic sets for _n for odd primes p.

Construction of p-basic sets for symmetric groups with special properties.

Results on decomposition numbers of _n.

Abstract

This article concerns the $p$-basic set existence problem in the representation theory of finite groups. We show that, for any odd prime $p$, the alternating group $\A_n$ has a $p$-basic set. More precisely, we prove that the symmetric group $\sym_n$ has a $p$-basic set with some additional properties, allowing us to deduce a $p$-basic set for $\A_n$. Our main tool is the generalized perfect isometries introduced by K\"ulshammer, Olsson and Robinson. As a consequence we obtain some results on the decomposition number of $\A_n$.