The integral cohomology rings of Peterson varieties in type A

TL;DR

This paper investigates the integral cohomology rings of Peterson varieties of type A, providing descriptions via symmetric invariants and explicit generators and relations, advancing understanding of their algebraic structure.

Contribution

It offers two novel descriptions of the cohomology ring of Peterson varieties, including an isomorphism with symmetric invariants and an explicit presentation with generators and relations.

Findings

Cohomology ring is isomorphic to symmetric invariants of permutohedral variety

Explicit generators and relations for the ring structure

Enhanced understanding of Peterson variety topology

Abstract

In this paper, we study the ring structure of the integral cohomology of the Peterson variety of type $\text{A}_{n-1}$. We give two kinds of descriptions: (1) we show that it is isomorphic to the $\mathfrak{S}_n$-invariant subring of the integral cohomology ring of the permutohedral variety, (2) we determine the ring structure in terms of ring generators and their relations.