The equivariant cohomology rings of Peterson varieties

Abstract

The main result of this note is an efficient presentation of the $S^1$-equivariant cohomology ring of Peterson varieties (in type $A$) as a quotient of a polynomial ring by an ideal $\mathcal{J}$, in the spirit of the well-known Borel presentation of the cohomology of the flag variety. Our result simplifies previous presentations given by Harada-Tymoczko and Bayegan-Harada. In particular, our result gives an affirmative answer to a conjecture of Bayegan and Harada that the defining ideal $\mathcal{J}$ is generated by quadratics.

14 pages. Minor changes made in convention and notation in order to be consistent with earlier papers in the literature

References in corpus (2)

• A positive Monk formula in the S^1-equivariant cohomology of type A Peterson varieties
• The cohomology ring of the GKM graph of a flag manifold of classical type

Cited by in corpus (5)

• Hessenberg varieties and hyperplane arrangements
• The cohomology rings of regular nilpotent Hessenberg varieties in Lie type A
• The torus equivariant cohomology rings of Springer varieties
• An additive basis for the cohomology rings of regular nilpotent Hessenberg varieties
• Coordinate rings of regular nilpotent Hessenberg varieties in the open opposite Schubert cell