Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Higher Specht bases and $q$-series for the cohomology rings of certain Hessenberg varieties

Kyle Salois

Abstract

It is conjectured (following the Stanley-Stembridge conjecture) that the cohomology rings of regular semisimple Hessenberg varieties yield permutation representations, but the decompositions of the modules are only known in some cases. For the Hessenberg function $h=(h(1),n,\ldots,n)$, the structure of the cohomology ring was determined by Abe, Horiguchi, and Masuda in 2017. We define two new bases for this cohomology ring, one of which is a higher Specht basis, and the other of which is a permutation basis. We also examine the transpose Hessenberg variety, indexed by the Hessenberg function $h' = ((n-1)^{n-m},n^m)$, and show that analogous results hold. Further, we give combinatorial bijections between the monomials in the new basis and sets of $P$-tableaux, motivated by the work of Gasharov, illustrating the connections between the $\mathfrak{S}_n$ action on these cohomology rings and the Schur expansion of chromatic symmetric functions.

Higher Specht bases and $q$-series for the cohomology rings of certain Hessenberg varieties

Abstract

It is conjectured (following the Stanley-Stembridge conjecture) that the cohomology rings of regular semisimple Hessenberg varieties yield permutation representations, but the decompositions of the modules are only known in some cases. For the Hessenberg function , the structure of the cohomology ring was determined by Abe, Horiguchi, and Masuda in 2017. We define two new bases for this cohomology ring, one of which is a higher Specht basis, and the other of which is a permutation basis. We also examine the transpose Hessenberg variety, indexed by the Hessenberg function , and show that analogous results hold. Further, we give combinatorial bijections between the monomials in the new basis and sets of -tableaux, motivated by the work of Gasharov, illustrating the connections between the action on these cohomology rings and the Schur expansion of chromatic symmetric functions.

Paper Structure

This paper contains 16 sections, 25 theorems, 61 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Acknowledgements
  4. Background
  5. Hessenberg Varieties
  6. Chromatic Symmetric Functions
  7. GKM Graphs
  8. Symmetric Group Representations
  9. Basis Elements when $h'=((n-1)^{n-m},n^m)$
  10. The Cohomology Basis for the Transpose Hessenberg Function
  11. Higher Specht Basis for $H^*(\mathrm{Hess}(S,h))$
  12. Permutation Representations
  13. Tableaux Bijections
  14. The Regular Nilpotent Case
  15. The Regular Semisimple Case
  16. ...and 1 more sections

Key Result

Theorem 1.1

The following sets form a basis of $H^{*}(\mathrm{Hess}(S,h))$ when $h=(h(1),n,\ldots,n)$: running over all $0\leq i_j \leq n-j$ in the first equation, and over all $0\leq \ell_j\leq n-1-j$, and $1\leq k\leq n-1$ in the second equation. We denote these sets of monomials $B_1$ and $B_3$.

Figures (5)

  • Figure 1: The (unlabeled) GKM graphs for two Hessenberg functions. Dashed lines represent the permutation $(13)$, solid lines represent $(12)$, and doubled lines represent $(23)$.
  • Figure 2: Two GKM classes for $h=(2,3,3)$, as defined in AHM.
  • Figure 3: Two GKM classes for $h=(2,3,3)$, as given in Definition \ref{['def:TransposeClasses']}.
  • Figure 4: The transition matrix between $B_1\cup B_2$ and $B_1\cup B_3$ for elements of degree $0$ when $h(1)=1$.
  • Figure 5: The general form of the transition matrix between $B_1\cup B_2$ and $B_1\cup B_3$.

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1: Ty05
  • Proposition 2.2
  • proof
  • Corollary 2.3
  • proof
  • ...and 43 more