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Regular semisimple Hessenberg varieties with cohomology rings generated in degree two

Mikiya Masuda, Takashi Sato

TL;DR

This work provides a complete characterization of when the cohomology ring $H^*(X(h))$ of a regular semisimple Hessenberg variety is generated in degree two. By leveraging GKM theory for $H^*_T(X(h))$ and a Morse-Bott moment-map framework, the authors prove a necessary condition, ruling out degree-two generation except for Hessenberg functions of a double lollipop form. They then establish sufficiency by showing that such $X(h)$ fibers over a toric base with a product of flag varieties as fiber, ensuring the degree-two generation from both base and fiber. The results connect geometric structure—via toric fibrations and explicit generators—to combinatorial patterns in Hessenberg functions, with implications for representation theory and chromatic symmetric functions via the Hà-Burghon-Shareshian–Wachs context.

Abstract

A regular semisimple Hessenberg variety $\mathrm{Hess}(S,h)$ is a smooth subvariety of the flag variety determined by a square matrix $S$ with distinct eigenvalues and a Hessenberg function $h$. The cohomology ring $H^*(\mathrm{Hess}(S,h))$ is independent of the choice of $S$ and is not explicitly described except for a few cases. In this paper, we characterize the Hessenberg function $h$ such that $H^*(\mathrm{Hess}(S,h))$ is generated in degree two as a ring. It turns out that such $h$ is what is called a (double) lollipop.

Regular semisimple Hessenberg varieties with cohomology rings generated in degree two

TL;DR

This work provides a complete characterization of when the cohomology ring of a regular semisimple Hessenberg variety is generated in degree two. By leveraging GKM theory for and a Morse-Bott moment-map framework, the authors prove a necessary condition, ruling out degree-two generation except for Hessenberg functions of a double lollipop form. They then establish sufficiency by showing that such fibers over a toric base with a product of flag varieties as fiber, ensuring the degree-two generation from both base and fiber. The results connect geometric structure—via toric fibrations and explicit generators—to combinatorial patterns in Hessenberg functions, with implications for representation theory and chromatic symmetric functions via the Hà-Burghon-Shareshian–Wachs context.

Abstract

A regular semisimple Hessenberg variety is a smooth subvariety of the flag variety determined by a square matrix with distinct eigenvalues and a Hessenberg function . The cohomology ring is independent of the choice of and is not explicitly described except for a few cases. In this paper, we characterize the Hessenberg function such that is generated in degree two as a ring. It turns out that such is what is called a (double) lollipop.
Paper Structure (8 sections, 13 theorems, 98 equations, 3 figures)

This paper contains 8 sections, 13 theorems, 98 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Regular semisimple Hessenberg varieties
  3. Necessity
  4. Moment maps
  5. The case $h=(2,n-1,\dots,n-1,n,n)$
  6. Poincaré polynomial of $X(h)$
  7. Hilbert series of the subring generated by $H^2(X(h))$
  8. Sufficiency

Key Result

Theorem 1.1

Assume that $h(j)\ge j+1$ for $j\in [n-1]$. Then $H^*(X(h))$ is generated in degree $2$ as a ring if and only if $h$ is of the following form eq:h for some $1\le a< b\le n$,

Figures (3)

  • Figure 1: The configurations for $h = (2,3,4,5,6,7,11,11,11,11)$ and $h = (4,4,4,5,6,7,11,11,11,11)$
  • Figure 2: The labeled graphs of $X(h)$ and $X(h')$
  • Figure 3: The configuration corresponding to $h^j$.

Theorems & Definitions (27)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 2.1: ma-pr-sh
  • Example 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.1
  • Theorem 2.4: AMS1
  • Remark 2.2: see Subsection 6.2 in AMS1 for more details
  • Example 2.2
  • ...and 17 more