Linear conditions imposed on flag varieties

2004/06/26 by Julianna S. Tymoczko, Julianna Tymoczko, Tymoczko, Julianna S. · 2 citations
Mathematics · #05E10 #14F25 #14M15 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #math.AG #math.CO #msc:05E10 #msc:14F25 #msc:14M15

paper · pdf · doi:10.48550/arxiv.math/0406541

16 pages; v2 and v3 contain editorial revisions and acknowledgements

openalex publication_date 2004/06/26 · arxiv created 2005/10/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study subvarieties of the flag variety called Hessenberg varieties, defined by certain linear conditions. These subvarieties arise naturally in applications including geometric representation theory, number theory, and numerical analysis. We describe completely the homology of Hessenberg varieties over GL_n(C) and show that they have no odd-dimensional homology. We provide an explicit geometric construction which partitions each Hessenberg variety into pieces homeomorphic to affine space. We characterize these affine pieces by fillings of Young tableaux and show that the dimension of the affine piece can be computed by combinatorial rules generalizing the Eulerian numbers. We give an equivalent formulation of this result in terms of roots. We conclude with a section on open questions.

Cited by

• Type A Nilpotent Hessenberg varieties with Hessenberg function h(i)≤ i+1
• Markov chains on Weyl groups from the geometry of the flag variety

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