Linear conditions imposed on flag varieties
By Julianna Tymoczko. 2004
96 citations recorded by the source corpus.
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.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/1db1bd5f414c525772b11967b3eb7fead265b458
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A (2015, background): “This map is surjective since Hess(N, h) admits a paving by complex affines [48] as mentioned in Section 2 and hence the Serre spectral sequence collapses at E2-stage [35, Chapter III, Theorem 2.”
- A positive Monk formula in the S1‐equivariant cohomology of type A Peterson varieties (2009, background): “Hessenberg varieties arise in many areas of mathematics, including geometric representation theory [12, 31, 33], numerical analysis [8], mathematical physics [24, 29], combinatorics [10], and algebraic geometry [5, 6, 34, 35].”
- The cohomology of abelian Hessenberg varieties and the Stanley–Stembridge conjecture (2017, background): “Note that if g ∈ GL(n,C), then Hess(X, h) and Hess(gXg−1, h) can be identified via the action of GL(n,C) on F`ags(C) [29].”