Hiraku Abe (Vol. 18 No. 2 2014)

The equivariant cohomology rings of regular nilpotent Hessenberg varieties in Lie type A: Research Announcement 
Hiraku Abe, Megumi Harada, Tatsuya Horiguchi, and Mikiya Masuda

Let n be a fixed positive integer and   a Hessenberg function. The main result of this manuscript is to give a systematic method for producing an explicit presentation by generators and relations of the equivariant and ordinary cohomology rings (with  coefficients) of any regular nilpotent Hessenberg variety Hess(h) in type A. Specifically, we give an explicit algorithm, depending only on the Hessenberg function h, which produces the n defining relations  in the equivariant cohomology ring. Our result generalizes known results: for the case h = (2, 3, 4, . . . , n, n), which corresponds to the Peterson variety  , Petn we recover the presentation of    given previously by Fukukawa, Harada, and Masuda. Moreover, in the case h = (n, n, . . . , n), for which the corresponding regular nilpotent Hessenberg variety is the full flag variety  , we can explicitly relate the generators of our ideal with those in the usual Borel presentation of the cohomology ring of  . The proof of our main theorem includes an argument that the restriction homomorphis   is surjective. In this research announcement, we briefly recount the context and state our results; we also give a sketch of our proofs and conclude with a brief discussion of open questions. A manuscript containing more details and full proofs is forthcoming.

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