Recursive sequences attached to modular representations of finite groups

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⊗n have algebraic Hilbert series when M is Omega-algebraic, in the sense that the non-projective summands of M⊗n fall into finitely many orbits under the action of the syzygy operator Ω. Similarly, we prove that these dimension sequences are eventually linearly recursive when M is what we term Ω+-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.