The Cohomology Rings of Regular Nilpotent Hessenberg Varieties in Lie Type A

Let n be a fixed positive integer and h: \1,2,…,n\ → \1,2,…,n\ a Hessenberg function. The main results of this paper are two-fold. First, we give a systematic method, depending in a simple manner on the Hessenberg function h, for producing an explicit presentation by generators and relations of the cohomology ring H⁽Hess(N,h)) with Q coefficients of the corresponding regular nilpotent Hessenberg variety Hess(N,h). Our result generalizes known results in special cases such as the Peterson variety and also allows us to answer a question posed by Mbirika and Tymoczko. Moreover, our list of generators in fact forms a regular sequence, allowing us to use techniques from commutative algebra in our arguments. Our second main result gives an isomorphism between the cohomology ring H^*(Hess(N,h)) of the regular nilpotent Hessenberg variety and the Sₙ-invariant subring H^*(Hess(S,h))Sₙ of the cohomology ring of the regular semisimple Hessenberg variety (with respect to the Sₙ-action on H^*(Hess(S,h)) defined by Tymoczko). Our second main result implies that dim_ Q Hᵏ(Hess(N,h)) = dim_ Q Hᵏ(Hess(S,h))Sₙ for all k and hence partially proves the Shareshian–Wachs conjecture in combinatorics, which is in turn related to the well-known Stanley–Stembridge conjecture. A proof of the full Shareshian–Wachs conjecture was recently given by Brosnan and Chow, and independently by Guay–Paquet, but in our special case, our methods yield a stronger result (i.e., an isomorphism of rings) by more elementary considerations. This article provides detailed proofs of results we recorded previously in a research announcement [2].