A Giambelli formula for the $S^1$-equivariant cohomology of type $A$ Peterson varieties

Abstract

We prove a Giambelli formula for the Peterson Schubert classes in the $S^1$-equivariant cohomology ring of a type $A$ Peterson variety. The proof uses the Monk formula for the equivariant structure constants for the Peterson Schubert classes derived by Harada and Tymoczko. In addition, we give proofs of two facts observed by H. Naruse: firstly, that some constants that appear in the multiplicative structure of the $S^1$-equivariant cohomology of Peterson varieties are Stirling numbers of the second kind, and secondly, that the Peterson Schubert classes satisfy a stability property in a sense analogous to the stability of the classical equivariant Schubert classes in the $T$-equivariant cohomology of the flag variety.