On Fixed Points of a Map Defined by Alain Connes

TL;DR

This paper investigates fixed points of a Connes-inspired map, characterizing non-zero fixed points as an abelian group, providing explicit examples, and revealing most are not one-dimensional representations, indicating ongoing research needs.

Contribution

It introduces a new analysis of fixed points of a Connes-based map, explicitly characterizing them and providing concrete examples.

Findings

Non-zero fixed points form an abelian group

Most fixed points are not one-dimensional representations

Explicit calculations of examples are provided

Abstract

In this paper we study certain analogues of the map defined by Alain Connes, which follows an idea of Atiyah in trying to simplify the proof of Feit-Thompson theorem. It turns out that the the non-zero fixed points of the map can be characterized explicitly as an abelian group, and we can calculate some examples. It turns out that most fixed points are not one dimensional representations for these examples, and much work need to be done.