Vol. 5, No. 2, 2012

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

Darius Bayegan and Megumi Harada

Vol. 5 (2012), No. 2, 115–132
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.

Keywords
Giambelli formula, Peterson variety, Schubert calculus, equivariant cohomology
Primary: 14N15
Secondary: 55N91
Milestones
Received: 2 January 2011
Revised: 12 September 2011
Accepted: 13 October 2011
Published: 27 January 2013

Communicated by Ravi Vakil
Authors
 Darius Bayegan Department of Pure Mathematics and Mathematical Statistics Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge CB30WA United Kingdom Megumi Harada Department of Mathematics and Statistics McMaster University 1280 Main Street, West Hamilton, Ontario L8S4K1 Canada