Recursive sequences attached to modular representations of finite groups

2021/05/11 by Alexandru Chirvăsitu, Chirvasitu, Alexandru, Tara Hudson +3 · 1 citation
Mathematics · #11K31 #13F25 #16D40 #20C05 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Representation Theory (math.RT) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2105.04732

openalex publication_date 2021/05/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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• Non-negligible summands in tensor powers of some modular representations of finite p-groups

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