[Paper Review] Commutative Banach algebras and modular representation theory

David J. Benson|arXiv (Cornell University)|Aug 30, 2020

Advanced Operator Algebra Research54 references4 citations

TL;DR

This paper develops a Banach algebra framework for modular representation theory by completing the complexified representation ring, enabling the application of Gelfand's spectral theory to study algebra homomorphisms to ℂ. A key result is that the Jacobson radical and nil radical of the representation ring always coincide, revealing a deep structural property of these algebras.

ABSTRACT

In a recent paper of Benson and Symonds, a new invariant was introduced for modular representations of a finite group. An interpretation was given as a spectral radius with respect to a Banach algebra completion of the representation ring. Our purpose here is to take these notions further, and investigate the structure of the resulting Banach algebras. Some of the material in that paper is repeated here in greater generality, and for clarity of exposition. We give an axiomatic definition of an abstract representation ring, and representation ideal. The completion is then a commutative Banach algebra, and the techniques of Gelfand from the 1940s are applied in order to study the space of algebra homomorphisms to $\mathbb C$. One surprising consequence of this investigation is that the Jacobson radical and the nil radical of a (complexified) representation ring always coincide. These notes are intended for representation theorists. So background material on commutative Banach algebras is given in detail, whereas representation theoretic background is more condensed.

Motivation & Objective

• To extend Benson and Symonds' invariant for modular representations using Banach algebra completion.
• To provide a systematic study of the resulting commutative Banach algebras arising from representation rings.
• To clarify the structure of the space of algebra homomorphisms to ℂ using Gelfand theory.
• To establish foundational results in commutative Banach algebras for representation theorists.
• To demonstrate that the Jacobson radical and nil radical of a complexified representation ring are always equal.

Proposed method

• Introduce an axiomatic framework for abstract representation rings and representation ideals.
• Construct a Banach algebra completion of the complexified representation ring using a spectral norm.
• Apply Gelfand's theory of commutative Banach algebras to analyze the space of continuous algebra homomorphisms to ℂ.
• Characterize the spectrum of the Banach algebra as the set of such homomorphisms.
• Use spectral radius formulas to interpret the invariant introduced by Benson and Symonds.
• Leverage the Gelfand transform and spectral permanence to deduce structural properties of the algebra.

Experimental results

Research questions

• RQ1 How can the representation ring of a finite group be completed into a commutative Banach algebra to facilitate spectral analysis?
• RQ2 What is the relationship between the Jacobson radical and the nil radical in the complexified representation ring?
• RQ3 How does the spectrum of the Banach algebra completion relate to the representation-theoretic data of the group?
• RQ4 In what way does the spectral radius of an element in the completion reflect modular representation-theoretic invariants?
• RQ5 What structural properties of the representation ring emerge from the application of Gelfand theory?

Key findings

• The Jacobson radical and the nil radical of the complexified representation ring coincide, indicating a strong structural constraint on the ring's ideal structure.
• The spectrum of the Banach algebra completion is identified with the space of algebra homomorphisms to ℂ, enabling a topological interpretation of representation-theoretic data.
• The spectral radius of an element in the completion corresponds to a representation-theoretic invariant, generalizing Benson and Symonds' construction.
• The Gelfand transform provides a concrete realization of the Banach algebra as an algebra of continuous functions on its spectrum.
• The completion process preserves essential representation-theoretic information while endowing the ring with a Banach algebra topology.
• The axiomatic framework allows for generalization beyond finite groups, applying to abstract representation rings with suitable ideals.