Minimum codimension of eigenspaces in irreducible representations of simple linear algebraic groups

2022/12/07 by Ana-M. Retegan, Retegan, Ana-M.
Mathematics · #20G05 (primary) #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2212.03643

openalex publication_date 2022/12/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let k be an algebraically closed field of characteristic p ≥ 0, let G be a simple simply connected classical linear algebraic group of rank ℓ and let T be a maximal torus in G with rational character group X(T). For a nonzero p-restricted dominant weight λ∈ X(T), let V be the associated irreducible kG-module. Define ν_G(V) to be the minimum codimension of eigenspaces corresponding to non-central elements of G on V. In this paper, we calculate ν_G(V) for G of type A_ℓ, ℓ ≥ 16, and dim(V) ≤ \fracℓ³2; for G of type B_ℓ, respectively C_ℓ, ℓ ≥ 14, and dim(V) ≤ 4ℓ³; and for G of type D_ℓ, ℓ ≥ 16, and dim(V) ≤ 4ℓ³. Moreover, for the groups of smaller rank and their corresponding irreducible modules with dimension satisfying the above bounds, we determine lower-bounds for ν_G(V).

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