University of Birmingham > Talks@bham > Algebra seminar  > Minimum eigenspace codimension in irreducible representations of simple classical linear algebraic groups

Minimum eigenspace codimension in irreducible representations of simple classical linear algebraic groups

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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 non-zero p-restricted dominant weight λ ∈ X(T), let V be the associated irreducible kG-module. Let Vg(μ) denote the eigenspace corresponding to the eigenvalue μ ∈ k of g ∈ G on V and define

νG(V)=min{dim(V) − dim(Vg(μ)) | g ∈ G \ Z(G), μ ∈ k}

to be the minimum eigenspace codimension on V. In this talk, we determine νG(V) for G of type Al, l ≥ 16 and dim(V) ≤l3/2; for G of type Bl, Cll ≥ 14 and dim(V) ≤ 4l3; and for G of type Dl, 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 νG(V).

This talk is part of the Algebra seminar series.

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