Abstract
In previous work we developed a general formalism for equivariant Schubert calculus of grassmannians consisting of a basis theorem, a Pieri formula and a Giambelli formula. Part of the work consists in interpreting the results in a ring that can be considered as the formal generalized analog of localized equivariant cohomology of infinite grassmannians. Here we present an extract of the theory containing the essential features of this ring. In particular we emphasize the importance of the GKM condition. Our formalism and methods are influenced by the combinatorial formalism given by A. Knutson and T. Tao for equivariant cohomology of grassmannians, and of the use of factorial Schur polynomials in the work of L.C. Mihalcea.
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References
M. Goresky, R. Kottwitz, & R. MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), 25–83.
A. Knutson & T. Tao, Puzzles and (equivariant) cohomology of Grassmannians, Duke Math. J. 119 (2003), 221–260.
D. Laksov, Schubert calculus and equivariant cohomology of Grassmannians, Advances in Math. 217 (2008), 1869–1888.
D. Laksov, A formalism for equivariant Schubert calculus, Algebra Number Theory 3 (2009), 711–727.
I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford Mathematical Monographs. Oxford Science Publications. Second edition. With contributions by A. Zelevinsky, The Clarendon Press, Oxford University Press, Springer, Oxford (1995).
L.C. Mihalcea, Equivariant quantum Schubert calculus, Adv. Math. 203 (2006), 1–33.
L.C. Mihalcea, Giambelli Formulae for the equivariant quantum cohomology of the Grassmannian, Trans. Amer. Math. Soc. 360 (2008), 2285–2301.
A. Molev & B.E. Sagan, A Littlewood–Richardson rule for factorial Schur functions, Trans. Amer. Math. 351 (1999), 4429–4443.
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Laksov, D. (2011). A Relation Between Symmetric Polynomials and the Algebra of Classes, Motivated by Equivariant Schubert Calculus. In: Fløystad, G., Johnsen, T., Knutsen, A. (eds) Combinatorial Aspects of Commutative Algebra and Algebraic Geometry. Abel Symposia, vol 6. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19492-4_7
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DOI: https://doi.org/10.1007/978-3-642-19492-4_7
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