Abstract. 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 twofold. 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 H k (Hess(N, h)) = dim Q H k (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, but in our special case, our methods yield a stronger result (i.e. an isomorphism of rings) by more elementary considerations. This paper provides detailed proofs of results we recorded previously in a research announcement.
We investigate the cohomology rings of regular semisimple Hessenberg varieties whose Hessenberg functions are of the form h = (h(1), n . . . , n) in Lie type A n−1 . The main result of this paper gives an explicit presentation of the cohomology rings in terms of generators and their relations. Our presentation naturally specializes to Borel's presentation of the cohomology ring of the flag variety and it is compatible with the representation of the symmetric group S n on the cohomology constructed by J. Tymoczko. As a corollary, we also give an explicit presentation of the S n -invariant subring of the cohomology ring.
Abstract. Let G be a complex semisimple linear algebraic group and let Pet be the associated Peterson variety in the flag variety G/B. The main theorem of this note gives an efficient presentation of the equivariant cohomology ring H * S (Pet) of the Peterson variety as a quotient of a polynomial ring by an ideal J generated by quadratic polynomials, in the spirit of the Borel presentation of the cohomology of the flag variety. Here the group S ∼ = C * is a certain subgroup of a maximal torus T of G. Our description of the ideal J uses the Cartan matrix and is uniform across Lie types. In our arguments we use the Monk formula and Giambelli formula for the equivariant cohomology rings of Peterson varieties for all Lie types, as obtained in the work of Drellich. Our result generalizes a previous theorem of Fukukawa-Harada-Masuda, which was only for Lie type A.
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