Recursive sequences attached to modular representations of finite groups

TL;DR

This paper studies the algebraic properties of the dimensions of cores of tensor powers of modular representations of finite groups, showing they have algebraic Hilbert series or are linearly recursive under certain conditions.

Contribution

It introduces the concepts of Omega-algebraic and Omega-plus-algebraic modules, proving algebraic Hilbert series and linear recursiveness of core dimensions, partially confirming Benson and Symonds' conjecture.

Findings

Dimensions have algebraic Hilbert series for Omega-algebraic modules.

Core dimension sequences are eventually linearly recursive for Omega-plus-algebraic modules.

Provides auxiliary results on linear recurrence properties of multi-variable sequences.

Abstract

The core of a finite-dimensional modular representation $M$ of a finite group $G$ is its largest non-projective summand. We prove that the dimensions of the cores of $M^{\otimes n}$ have algebraic Hilbert series when $M$ is Omega-algebraic, in the sense that the non-projective summands of $M^{\otimes n}$ fall into finitely many orbits under the action of the syzygy operator $\Omega$. Similarly, we prove that these dimension sequences are eventually linearly recursive when $M$ is what we term $\Omega^{+}$-algebraic. This partially answers a conjecture by Benson and Symonds. Along the way, we also prove a number of auxiliary permanence results for linear recurrence under operations on multi-variable sequences.