The integral cohomology rings of Peterson varieties in type A

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.