K-theory Schubert calculus of the affine Grassmannian

By Thomas Lam, Anne Schilling, Mark Shimozono. Compositio Mathematica, 2009

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Abstract We construct the Schubert basis of the torus-equivariant K-homology of the affine Grassmannian of a simple algebraic group G, using the K-theoretic NilHecke ring of Kostant and Kumar. This is the K-theoretic analogue of a construction of Peterson in equivariant homology. For the case where G=SLn, the K-homology of the affine Grassmannian is identified with a sub-Hopf algebra of the ring of symmetric functions. The Schubert basis is represented by inhomogeneous symmetric functions, calledK-k-Schur functions, whose highest-degree term is a k-Schur function. The dual basis in K-cohomology is given by the affine stable Grothendieck polynomials, verifying a conjecture of Lam. In addition, we give a Pieri rule in K-homology. Many of our constructions have geometric interpretations by means of Kashiwara’s thick affine flag manifold.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7866750b2bc65cf27541286b53a2597b07a96bc2

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Citations in the corpus

k-Schur Functions and Affine Schubert Calculus (2013, methodology): “Chapter 4 is Mark Shimozono’s synopsis of his lectures on “generalizations to other affine group types”.”