[Paper Review] On the Burnside-Brauer-Steinberg theorem

Benjamin Steinberg|arXiv (Cornell University)|Sep 26, 2014

Algebraic structures and combinatorial models10 references3 citations

TL;DR

This paper establishes the first known analogue of Brauer's refinement of Burnside's theorem for finite monoids, proving that if a faithful representation of a finite monoid has a character taking r distinct values, then all irreducible representations appear as composition factors in the first r tensor powers. It further proves a similar bound for symmetric powers using characteristic polynomials of operators, extending classical results from group representation theory to monoids.

ABSTRACT

A well-known theorem of Burnside says that if $ρ$ is a faithful representation of a finite group $G$ over a field of characteristic $0$, then every irreducible representation of $G$ appears as a constituent of a tensor power of $ρ$. In 1962, R. Steinberg gave a module theoretic proof that simultaneously removed the constraint on the characteristic, and allowed the group to be replaced by a monoid. Brauer subsequently simplified Burnside's proof and, moreover, showed that if the character of $ρ$ takes on $r$ distinct values, then the first $r$ tensor powers of $ρ$ already contain amongst them all of the irreducible representations of $G$ as constituents. In this note we prove the analogue of Brauer's result for finite monoids. We also prove the corresponding result for the symmetric powers of a faithful representation.

Motivation & Objective

• To extend Brauer's refinement of Burnside's theorem from finite groups to finite monoids.
• To determine a uniform bound on the number of tensor powers required to realize all irreducible representations of a finite monoid.
• To establish an analogous bound for symmetric powers of faithful representations in terms of the number of distinct characteristic polynomials of the representation matrices.
• To demonstrate that the minimal faithful module generated by symmetric powers cannot be bounded solely by the number of distinct characteristic polynomials, unlike in the group case.

Proposed method

• Uses primitive idempotents in the monoid algebra to characterize when an irreducible representation appears as a composition factor.
• Applies generating functions of symmetric power characters to derive rational functions with bounded denominator degree.
• Employs Molien-type generating functions to relate the character of symmetric powers to the characteristic polynomials of operators.
• Leverages the structure theory of irreducible representations of finite monoids, particularly the role of apex idempotents.
• Uses Newton's identities to relate character values to characteristic polynomials, enabling bounds on the number of distinct polynomials.
• Applies the fact that the character of a faithful representation on a monoid's idempotent apex is non-zero to ensure non-vanishing generating functions.

Experimental results

Research questions

• RQ1 Can Brauer’s bound on the number of tensor powers needed to realize all irreducible representations be extended from finite groups to finite monoids?
• RQ2 Is there a uniform bound on the number of symmetric powers required to realize all irreducible representations of a finite monoid, in terms of the number of distinct characteristic polynomials of the representation matrices?
• RQ3 Does the minimal k such that the direct sum of symmetric powers up to k is faithful depend only on the number of distinct characteristic polynomials, or are additional invariants needed?
• RQ4 How does the failure of orthogonality in monoid character theory affect the applicability of Brauer’s original character-theoretic method?

Key findings

• For a finite monoid M and a faithful representation ρ over a field of characteristic 0, if the character of ρ takes on r distinct values, then every irreducible representation of KM appears as a composition factor in one of the first r tensor powers V^{igotimes i} for i = 0, ..., r-1.
• The number of symmetric powers needed to realize all irreducible representations is bounded by dim(V) · s, where s is the number of distinct characteristic polynomials of the operators ρ(m) for m ∈ M.
• The generating function for the dimensions of symmetric powers of the image of a primitive idempotent is a non-zero rational function with denominator degree at most r, ensuring a non-zero coefficient within the first r terms.
• The bound r for tensor powers is sharp in the sense that it depends only on the number of distinct character values, not on the size of the monoid.
• The minimal k such that ⊕_{i=0}^k S^i(V) is a faithful KM-module cannot be bounded solely by dim(V) and the number of distinct characteristic polynomials, unlike in the group case.
• The proof relies on the non-vanishing of the coefficient of the apex idempotent in the group algebra, which ensures the generating function is non-zero and satisfies a linear recurrence of order r.