On an idea of Michael Atiyah

TL;DR

This paper explores Atiyah's idea of using iterative transformations of group representations, involving exterior powers and Adams operations, to approach the Feit-Thompson theorem on finite groups of odd order.

Contribution

It demonstrates that Atiyah's iterative approach extends to virtual representations and connects to exponential and Adams operations, providing a new perspective on the theorem.

Findings

G has odd order iff the transformation extends to virtual representations.

The transformation can be expressed via exponential and Adams operations.

Convergence to a non-trivial character illustrates the idea's relevance.

Abstract

In this note we investigate the idea of Michael Atiyah of using, as a possible approach to the Theorem of Feit-Thompson on the solvability of finite groups of odd order, the iterations of the transformation which replaces a representation of a finite group G on a finite dimensional complex vector space E by the difference between the associated representation of G on the sum of exterior powers of E and the trivial representation. We show that G has odd order if and only if the above operation extends to virtual representations and we then express it in terms of the exponential and the Adams operations in the complexified representation ring. We show the relevance of the idea in a concrete example by exhibiting convergence to a non-trivial character.