Anchors of irreducible characters

TL;DR

This paper introduces the concept of anchors of irreducible characters in finite groups, exploring their properties and relationships with other $p$-subgroups like defect groups and vertices, expanding understanding in modular representation theory.

Contribution

It defines and studies the properties of anchors of irreducible characters, relating them to existing $p$-subgroup invariants in finite group representation theory.

Findings

Anchors are uniquely associated with irreducible characters.

Properties of anchors are established and analyzed.

Connections between anchors, defect groups, and vertices are explored.

Abstract

Given a prime number $p$, every irreducible character $\chi$ of a finite group $G$ determines a unique conjugacy class of $p$-subgroups of $G$ which we will call the anchors of $\chi$. This invariant has been considered by L. Barker in the context of finite $p$-solvable groups. Besides proving some basic properties of these anchors, we investigate the relation to other $p$-groups which can be attached to irreducible characters, such as defect groups, vertices in the sense of J. A. Green and vertices in the sense of G. Navarro.