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

2025/08/21 by Vashaw, Kent B., Zhang, Justin · 1 citation
#18M05 #20C20 #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2508.15730

Abstract

Let p>0 be a prime, G be a finite p-group and \Bbbk be an algebraically closed field of characteristic p. Dave Benson has conjectured that if p=2 and V is an odd-dimensional indecomposable representation of G then all summands of the tensor product V ⊗ V^* except for \Bbbk have even dimension. It is known that the analogous result for general p is false. In this paper, we investigate the class of graded representations V which have dimension coprime to p and for which V ⊗ V^* has a non-trivial summand of dimension coprime to p, for a graded group scheme closely related to ℤ/p^r ℤ × ℤ/p^s ℤ, where r and s are nonnegative integers and p>2. We produce an infinite family of such representations in characteristic 3 and show in particular that the tensor subcategory generated by any of these representations in the semisimplification contains the modulo 3 reduction of the category of representations of the symmetric group S₃. Our results are compatible with a general version of Benson's conjecture due to Etingof.

Citations

• Fractal behavior of tensor powers of the two dimensional space in prime characteristic
• Asymptotic properties of tensor powers in symmetric tensor categories
• On the decomposition of tensor products of monomial modules for finite 2-groups
• On Frobenius exact symmetric tensor categories
• Recursive sequences attached to modular representations of finite groups
• The Benson -- Symonds Invariant for Ordinary and Signed Permutation Modules
• The Benson-Symonds Invariant for Permutation Modules
• New incompressible symmetric tensor categories in positive characteristic
• On symmetric fusion categories in positive characteristic
• Reductions of tensor categories modulo primes
• Endotrivial Modules for Finite Group Schemes
• Introduction to representation theory

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• Growth Problems of Quantum Groups

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