An explicit formula for the Hilbert series of a partial flag variety

2023/10/17 by Wayne A. Johnson, Johnson, Wayne A.
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2310.10932

openalex publication_date 2023/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a semisimple, simply-connected linear algebraic group, G, and parabolic subgroup, P⊆ G, we use the fact that the Hilbert polynomial of the equivariant embedding of G/P is equal to the Hilbert function to compute an explicit formula for the Hilbert series of G/P in terms of the dimensions of finitely many irreducible representations of G. As an example, we compute the Hilbert series of the adjoint variety of SL(n+1,ℂ). We conclude by computing the linear term of the numerator of the Hilbert series of any fundamental representation in type A.

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