Multiplicity free induction for the pairs (GL₂\timesGL₂,diag(GL₂)) and (SL₃,GL₂) over finite fields

2025/07/14 by Elias Depuydt, Depuydt, Elias, Maarten van Pruijssen +1
Computer Science · Mathematics · #20G40 (primary) 43A90 (secondary) #Advanced Algebra and Geometry #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2507.10790

openalex publication_date 2025/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We classify the irredible representations of GL₂(q) for which the induction to the product group GL₂(q)\timesGL₂(q), under the diagonal embedding, decomposes multiplicity free. It turns out that only the irreducible representations of dimensions 1 and q-1 have this property. We show that for GL₂(q) embedded into SL₃(q) via g\mapstodiag(g,det g^-1) none of the irreducible representations of GL₂(q) induce multiplicity free. In contrast, over the complex numbers, the holomorphic representation theory of these pairs is multiplicity free and the corresponding matrix coefficients are encoded by vector-valued Jacobi polynomials. We show that similar results cannot be expected in the context of finite fields for these examples.

Citations

• Multiplicity-free induced characters of symmetric groups

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