Growth Problems for Representations of Finite Monoids

TL;DR

The paper extends asymptotic growth results for tensor powers from groups to finite monoids by conjecturing that the indecomposable-summand growth rate equals that of the unit group restriction, and by deriving a Brauer-character–based formula for the leading term. It proves the conjecture under a injective-projective extension hypothesis and develops exact and asymptotic formulas for the length $l(n)$ in the nonsemisimple and semisimple regimes, with detailed computations for the full transformation monoid, the symmetric inverse monoid, and $ ext{M}(2,q)$. The approach relies on fusion graphs, decomposition matrices, and character theory to relate $b(n)$ and $l(n)$ to unit-group representations, and it provides explicit examples and code to compute the growth rates. These results illuminate how monoid structure governs tensor-power growth and offer practical tools for calculating asymptotics in concrete monoids relevant to algebra and combinatorics.

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.

Figures (6)

• Figure 1: The fusion graph and action matrix for $V=Y\oplus X$. The vertex colored in cyan is the null representation. In this case $a(n)=3^n$.
• Figure 2: The fusion graph and action matrix for the 12-dimensional projective $kT_4$-module in characteristic 0. The modules are labelled with their dimensions, and the projective cell $\Gamma^G$ is coloured in cyan.
• Figure 3: The ratio $b(n)/a(n)$ for the 12-dimensional projective $kT_4$-module in characteristic 0. Here $a(n)=5/12\cdot 12^n$ and $\lambda^{\textrm{sec}}=2$.
• Figure 4: The fusion graph and action matrix for the four-dimensional projective $k\mathop{\mathrm{M}}\nolimits(2,3)$-module in characteristic 3. The modules are labelled with their dimensions, and the projective cell $\Gamma^G$ is coloured in cyan.
• Figure 5: The ratio $b(n)/a(n)$ for the four-dimensional projective $k\mathop{\mathrm{M}}\nolimits(2,3)$-module in characteristic 3. Here $a(n)=(1/4+1/12(-1)^n)\cdot 4^n$ and $\lambda^{\text{sec}} = 1.$

Theorems & Definitions (22)

• Conjecture 1
• Remark 1
• Remark 2
• Proposition 1
• proof
• Example 1
• Theorem 1
• proof
• Remark 3
• Lemma 1