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Computations on the tautological basis of the cohomology ring of the Peterson variety

Yuito Hashimoto

TL;DR

The paper addresses the problem of computing square-free expansions for products of tautological degree-2 classes in the cohomology ring $H^*(Y)$ of the Peterson variety $Y$, a regular nilpotent Hessenberg variety. It develops a precise expansion for consecutive-index monomials $x_c(x_a\cdots x_b)$ using elementary symmetric polynomials and binomial coefficients, then extends to general monomials with a single square by introducing higher-order polynomials $e_k^{(l_1,...,l_s)}$ and handling the elimination of $x_n$ to stay within the tautological basis. The results establish integer-valued structure constants and provide concrete computational formulas, while open questions remain about direct formulas for $c_{J,K}^L$ and their geometric meaning. Overall, the work advances explicit combinatorial control over the cohomology ring structure of the Peterson variety and related Hessenberg varieties.

Abstract

It is known that the set of square free monomials on the Chern classes of the tautological line bundles over the Peterson variety forms an additive basis of its cohomology ring. We study the expansion formula for their products. In particular, we give a square free expansion of the products multiplying degree 2 classes in terms of elementary symmetric polynomials and binomial coefficients.

Computations on the tautological basis of the cohomology ring of the Peterson variety

TL;DR

The paper addresses the problem of computing square-free expansions for products of tautological degree-2 classes in the cohomology ring of the Peterson variety , a regular nilpotent Hessenberg variety. It develops a precise expansion for consecutive-index monomials using elementary symmetric polynomials and binomial coefficients, then extends to general monomials with a single square by introducing higher-order polynomials and handling the elimination of to stay within the tautological basis. The results establish integer-valued structure constants and provide concrete computational formulas, while open questions remain about direct formulas for and their geometric meaning. Overall, the work advances explicit combinatorial control over the cohomology ring structure of the Peterson variety and related Hessenberg varieties.

Abstract

It is known that the set of square free monomials on the Chern classes of the tautological line bundles over the Peterson variety forms an additive basis of its cohomology ring. We study the expansion formula for their products. In particular, we give a square free expansion of the products multiplying degree 2 classes in terms of elementary symmetric polynomials and binomial coefficients.
Paper Structure (6 sections, 19 theorems, 119 equations)

This paper contains 6 sections, 19 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Consecutive case
  4. General case
  5. Open questions
  6. Appendix

Key Result

Proposition 2.1

(a) There is a ring isomorphism which sends $($the quotient image of $)$$\varpi_k$ to $p_k\in H^{2}(\mathcal{Y})$, where $J_{\varpi}$ is the ideal generated by where we set $\varpi_0=\varpi_n=0$.

Theorems & Definitions (41)

  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Corollary 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Example 2.7
  • Lemma 2.8
  • Example 2.9
  • Lemma 3.1
  • ...and 31 more