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

We prove a Giambelli formula for the Peterson Schubert classes in the S1-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 S1-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.

Giambelli formula, Peterson variety, Schubert calculus, equivariant cohomology
Mathematical Subject Classification 2010
Primary: 14N15
Secondary: 55N91
Received: 2 January 2011
Revised: 12 September 2011
Accepted: 13 October 2011
Published: 27 January 2013

Communicated by Ravi Vakil
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