On ideals in group algebras: an uncertainty principle and the Schur product

TL;DR

This paper explores properties of ideals in finite group algebras, establishing a link between their dimension, minimal Hamming distance, and group order, and introduces the Schur product of ideals with structural insights.

Contribution

It generalizes an uncertainty principle for ideals in group algebras and introduces the Schur product, analyzing their module structure and conditions for ideals to coincide with their Schur square.

Findings

Established a link between ideal dimension, Hamming distance, and group order.

Introduced the Schur product of ideals and studied its properties.

Provided conditions for ideals to be equal to their Schur square, especially in p-group cases.

Abstract

In this paper we investigate some properties of ideals in group algebras of finite groups over fields. First, we highlight an important link between their dimension, their minimal Hamming distance and the group order. This is a generalized version of an uncertainty principle shown in 1992 by Meshulam. Secondly, we introduce the notion of the Schur product of ideals in group algebras and investigate the module structure and the dimension of the Schur square. We give a structural result on ideals that coincide with their Schur square, and we provide conditions for an ideal to be such that its Schur square has the projective cover of the trivial module as a direct summand. This has particularly interesting consequences for group algebras of p-groups over fields of characteristic p.