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Birational geometry of generalized Hessenberg varieties and the generalized Shareshian-Wachs conjecture

Young-Hoon Kiem, Donggun Lee

Abstract

We introduce generalized Hessenberg varieties and establish basic facts. We show that the Tymoczko action of the symmetric group $S_n$ on the cohomology of Hessenberg varieties extends to generalized Hessenberg varieties and that natural morphisms among them preserve the action. By analyzing natural morphisms and birational maps among generalized Hessenberg varieties, we give an elementary proof of the Shareshian-Wachs conjecture. Moreover we present a natural generalization of the Shareshian-Wachs conjecture that involves generalized Hessenberg varieties and provide an elementary proof. As a byproduct, we propose a generalized Stanley-Stembridge conjecture for weighted graphs. Our investigation into the birational geometry of generalized Hessenberg varieties enables us to modify them into much simpler varieties like projective spaces or permutohedral varieties by explicit sequences of blowups or projective bundle maps. Using this, we provide two algorithms to compute the $S_n$-representations on the cohomology of generalized Hessenberg varieties. As an application, we compute representations on the low degree cohomology of some Hessenberg varieties.

Birational geometry of generalized Hessenberg varieties and the generalized Shareshian-Wachs conjecture

Abstract

We introduce generalized Hessenberg varieties and establish basic facts. We show that the Tymoczko action of the symmetric group on the cohomology of Hessenberg varieties extends to generalized Hessenberg varieties and that natural morphisms among them preserve the action. By analyzing natural morphisms and birational maps among generalized Hessenberg varieties, we give an elementary proof of the Shareshian-Wachs conjecture. Moreover we present a natural generalization of the Shareshian-Wachs conjecture that involves generalized Hessenberg varieties and provide an elementary proof. As a byproduct, we propose a generalized Stanley-Stembridge conjecture for weighted graphs. Our investigation into the birational geometry of generalized Hessenberg varieties enables us to modify them into much simpler varieties like projective spaces or permutohedral varieties by explicit sequences of blowups or projective bundle maps. Using this, we provide two algorithms to compute the -representations on the cohomology of generalized Hessenberg varieties. As an application, we compute representations on the low degree cohomology of some Hessenberg varieties.
Paper Structure (18 sections, 28 theorems, 172 equations)

This paper contains 18 sections, 28 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Generalized Hessenberg varieties
  3. Definition and basic properties
  4. Torus action and the Tymoczko action
  5. Comparison maps and equivariance
  6. Birational geometry of generalized Hessenberg varieties
  7. Deletion and blowup
  8. An elementary proof of the Shareshian-Wachs conjecture
  9. Modifications
  10. The generalized Shareshian-Wachs conjecture
  11. New algorithms for the $S_n$-characters
  12. Algorithm 1: reduction to the permutohedral varieties
  13. Algorithm 2: reduction to the projective spaces
  14. Multiplicities of trivial representations
  15. Degree $k+1$ for $h_k$
  16. ...and 3 more sections

Key Result

Theorem 2.3

(1) For $h\in {\mathcal{H}}_{I,n}$ with $I$ in 27, the Hessenberg variety $X_h$ is a smooth projective variety of dimension (2) If $h(i)>i$ for all $i\in I$, $X_h$ is irreducible. If $h(i)=i$ for some $i\in I$, $X_h$ is isomorphic to the disjoint union of $\binom{n}{i}$ copies of $X_{h'}\times X_{h"}$ where (3) The cycle class map $A^*(X_h)\to H^*(X_h)$ from the Chow ring to the cohomology ring

Theorems & Definitions (78)

  • Definition 2.1
  • Example 2.2
  • Theorem 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • proof
  • Remark 2.7
  • Definition 2.8
  • Proposition 2.9
  • ...and 68 more