Representations and binomial coefficients

2023/10/12 by Peter Fiebig, Fiebig, Peter
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2310.08090

openalex publication_date 2023/10/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a root system R, a field K and a "choice of coefficients in K" we define a category of graded spaces with operators and study some of its properties. Then we assume that the coefficients are given by quantum binomials. We use basic arithmetic properties of binomial coefficients (such as q-versions of Lucas' theorem and the Pfaff-Saalschütz identity) to construct a Frobenius pull back functor and prove a Steinberg's tensor product formula. Then we show that these results imply the corresponding results for reductive algebraic groups in positive characteristics and for quantum groups at roots of unity.

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