Growth Problems for Representations of Finite Monoids

TL;DR

This paper proposes a conjecture on the asymptotic growth of indecomposable summands in tensor powers of finite monoid representations, with proofs under certain conditions and explicit formulas for specific monoids.

Contribution

It introduces a new conjecture relating growth rates to the monoid's character table and provides formulas and computational tools for key examples.

Findings

Conjecture relating growth rate to the monoid's character table

Proved the conjecture under an additional hypothesis

Computed growth rates for specific monoids like the full transformation monoid

Abstract

We give a conjecture for the asymptotic growth rate of the number of indecomposable summands in the tensor powers of representations of finite monoids, expressing it in terms of the (Brauer) character table of the monoid's group of units. We prove it under an additional hypothesis. We also give (exact and asymptotic) formulas for the growth rate of the length of the tensor powers when working over a good characteristic. As examples, we compute the growth rates for the full transformation monoid, the symmetric inverse monoid, and the monoid of 2 by 2 matrices. We also provide code used for our calculation.