The equivariant cohomology rings of Peterson varieties

The main result of this note gives an efficient presentation of the S¹-equivariant cohomology ring of Peterson varieties (in type A) as a quotient of a polynomial ring by an ideal 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 J is generated by quadratics.