Clebsch-Gordan and the theta filtration for modular representations of GL₂(\mathbb F_q)

2025/11/04 by Bhattacharjee, Srijeet, Ghate, Eknath, Pandey, Shivansh +1
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2511.02299

openalex publication_date 2025/11/04 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28

Abstract

Let p be a prime. We solve two problems in the mod p representation theory of GL₂(\mathbbF_q) where q=p^f. We first prove a Clebsch-Gordan decomposition theorem for the tensor product of two mod p representations of GL₂(\mathbbF_q). As an application, we use this to guess the structure of quotients of symmetric power representations of GL₂(\mathbbF_q) by submodules in the theta filtration. We then give a direct proof of this structure showing that such quotients are built out of principal series representations.

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