A positive Monk formula in the S^1-equivariant cohomology of type A Peterson varieties

摘要

Peterson varieties are a special class of Hessenberg varieties that have beenextensively studied e.g. by Peterson, Kostant, and Rietsch, in connection withthe quantum cohomology of the flag variety. In this manuscript, we develop ageneralized Schubert calculus, and in particular a positive Chevalley-Monkformula, for the ordinary and Borel-equivariant cohomology of the Petersonvariety $Y$ in type $A_{n-1}$, with respect to a natural $S^1$-action arisingfrom the standard action of the maximal torus on flag varieties. As far as weknow, this is the first example of positive Schubert calculus beyond the realmof Kac-Moody flag varieties $G/P$. Our main results are as follows. First, we identify a computationallyconvenient basis of $H^*_{S^1}(Y)$, which we call the basis of PetersonSchubert classes. Second, we derive a manifestly positive, integralChevalley-Monk formula for the product of a cohomology-degree-2 PetersonSchubert class with an arbitrary Peterson Schubert class. Both $H^*_{S^1}(Y)$and $H^*(Y)$ are generated in degree 2. Finally, by using our Chevalley-Monkformula we give explicit descriptions (via generators and relations) of boththe $S^1$-equivariant cohomology ring $H^*_{S^1}(Y)$ and the ordinarycohomology ring $H^*(Y)$ of the type $A_{n-1}$ Peterson variety. Our methodsare both directly from and inspired by those of GKM(Goresky-Kottwitz-MacPherson) theory and classical Schubert calculus. Wediscuss several open questions and directions for future work.