Capelli elements of the group algebra

TL;DR

This paper introduces Capelli elements derived from modified Capelli identities for irreducible representations, providing a basis for the center of the group algebra of finite groups.

Contribution

It develops a new approach to constructing a basis of the center of the group algebra using Capelli identities for irreducible representations.

Findings

Capelli elements form a basis of the center of the group algebra

Capelli identities are adapted for irreducible representations

Provides a new algebraic framework for understanding group algebra centers

Abstract

Inspired by the Capelli identities for group determinants obtained by T\^oru Umeda, we give a basis of the center of the group algebra of any finite group by using Capelli identities for irreducible representations. The Capelli identities for irreducible representations are modifications of the Capelli identity. These identities lead to Capelli elements of the group algebra. These elements construct a basis of the center of the group algebra.