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Notes on the cohomology of partial Hessenberg varieties

Tatsuya Horiguchi, Mikiya Masuda, Takashi Sato, Haozhi Zeng

TL;DR

This work studies the cohomology of partial Hessenberg varieties Hess$_{\Theta}(\mathsf{x},H)$ inside G/P by relating it to Hess$(\mathsf{x},H)$ in the full flag variety via a P/B fiber, enabling a Leray-Hirsch type decomposition $H^*(\mathrm{Hess}(\mathsf{x},H))\cong H^*(P/B)\otimes H^*(\mathrm{Hess}_{\Theta}(\mathsf{x},H))$ and an invariant description $H^*(\mathrm{Hess}_{\Theta}(\mathsf{x},H))\cong H^*(\mathrm{Hess}(\mathsf{x},H))^{W_{\Theta}(\text{star})}$. The authors develop the star action of $W_{\Theta}$ on cohomology, connect it with Borel’s presentation of $H^*(G/B)$, and establish the equivariant (and dot) actions in the regular semisimple setting, showing how these symmetries control the decomposition and invariant subspaces. They extend results to regular partial Hessenberg varieties, proving isomorphisms with invariants under dot actions and deriving duality properties, hard Lefschetz, and Hodge-Riemann relations in suitable cases. The paper also provides explicit Poincaré polynomials for toric cases via weight polytopes and, in type A, computes the double lollipop regular semisimple case, illustrating the utility of the fibration approach for concrete cohomological calculations. Overall, the work unifies geometric, combinatorial, and representation-theoretic perspectives to understand Hessenberg cohomology across full and partial flag settings.

Abstract

Hessenberg varieties are a family of subvarieties of full flag varieties. This family contains well-known varieties such as Springer fibers, Peterson varieties, and permutohedral varieties. It was introduced by De Mari-Procesi-Shayman in 1992 and has been actively studied in this decade. In particular, unexpected relations to hyperplane arrangements and the Stanley-Stembridge conjecture in graph theory have been discovered. Hessenberg varieties can be defined in partial flag varieties. In this paper, we study their cohomology by relating them to the cohomology of Hessenberg varieties in the full flag varieties.

Notes on the cohomology of partial Hessenberg varieties

TL;DR

This work studies the cohomology of partial Hessenberg varieties Hess inside G/P by relating it to Hess in the full flag variety via a P/B fiber, enabling a Leray-Hirsch type decomposition and an invariant description . The authors develop the star action of on cohomology, connect it with Borel’s presentation of , and establish the equivariant (and dot) actions in the regular semisimple setting, showing how these symmetries control the decomposition and invariant subspaces. They extend results to regular partial Hessenberg varieties, proving isomorphisms with invariants under dot actions and deriving duality properties, hard Lefschetz, and Hodge-Riemann relations in suitable cases. The paper also provides explicit Poincaré polynomials for toric cases via weight polytopes and, in type A, computes the double lollipop regular semisimple case, illustrating the utility of the fibration approach for concrete cohomological calculations. Overall, the work unifies geometric, combinatorial, and representation-theoretic perspectives to understand Hessenberg cohomology across full and partial flag settings.

Abstract

Hessenberg varieties are a family of subvarieties of full flag varieties. This family contains well-known varieties such as Springer fibers, Peterson varieties, and permutohedral varieties. It was introduced by De Mari-Procesi-Shayman in 1992 and has been actively studied in this decade. In particular, unexpected relations to hyperplane arrangements and the Stanley-Stembridge conjecture in graph theory have been discovered. Hessenberg varieties can be defined in partial flag varieties. In this paper, we study their cohomology by relating them to the cohomology of Hessenberg varieties in the full flag varieties.

Paper Structure

This paper contains 15 sections, 21 theorems, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Partial Hessenberg varieties
  3. Cohomology rings of partial Hessenberg varieties and Hessenberg varieties
  4. Star action
  5. Borel's presentation
  6. Main theorem
  7. Equivariant cohomology of regular semisimple partial Hessenberg varieties
  8. Regular semisimple Hessenberg varieties
  9. Regular semisimple partial Hessenberg varieties
  10. Star actions on equivariant cohomology
  11. Dot actions on equivariant cohomology
  12. Cohomology rings of regular partial Hessenberg varieties
  13. Poincaré polynomial
  14. Toric varieties in $G/P$
  15. Regular semisimple Hessenberg varieties for double lollipop case in type $A$

Key Result

Theorem 1.1

For any $\mathsf{x}\in \mathfrak{g}$ and any $\mathfrak{p}$-Hessenberg space $H$, the following hold: Here $H^*(\mathop{\mathrm{Hess}}\nolimits(\mathsf{x},H))^{W_\Theta({\rm star})}$ denotes the invariants in $H^*(\mathop{\mathrm{Hess}}\nolimits(\mathsf{x},H))$ under the star action of $W_\Theta$.

Figures (1)

  • Figure 1: The Dynkin diagram of type $A_{n-1}$ with vertices $\bullet$ in $\Theta = \Delta \setminus \{\alpha_i \mid a \leq i \leq b \}$.

Theorems & Definitions (40)

  • Theorem 1.1: Theorem \ref{['theorem:main']}
  • Proposition 1.2: Proposition \ref{['proposition:dotaction starinvariants']}
  • Theorem 1.3: Theorem \ref{['theorem:main2']}
  • Remark 2.1
  • Lemma 2.2
  • Remark 2.3
  • Example 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 30 more