2006/01/27 by Julianna S. Tymoczko, Julianna Tymoczko, Tymoczko, Julianna S.
Mathematics · #05E99 #14M15 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #math.AG #math.CO #msc:05E99 #msc:14M15
paper · pdf · doi:10.48550/arxiv.math/0601681
13 pages; corrected typos
openalex publication_date 2006/01/27 · arxiv created 2006/04/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a family of subvarieties of the flag variety defined by certain linear conditions, called Hessenberg varieties. We compare them to Schubert varieties. We prove that some Schubert varieties can be realized as Hessenberg varieties and vice versa. Our proof explicitly identifies these Schubert varieties by their permutation and computes their dimension. We use this to answer an open question by proving that Hessenberg varieties are not always pure dimensional. We give examples that neither semisimple nor nilpotent Hessenberg varieties need be pure; the latter are connected, non-pure-dimensional Hessenberg varieties. Our methods require us to generalize the definition of Hessenberg varieties.