K-theoretic analogue of factorial Schur P- and Q-functions

Details K-theoretic analogue of factorial Schur P- and Q-functions K-theoretic analogue of factorial Schur P- and Q-functions

2011-12-22

arxiv.org/abs/1112.5223v2

Mathematics

Combinatorics

39 pages, minor changes, acknowledgments added

Scientific paper

We introduce a $K$-theoretic analogue of factorial Schur $P$- and $Q$- functions. These functions are non-homogeneous symmetric functions generalizing Ivanov's factorial Schur $P$- and $Q$- functions. We prove various combinatorial expressions for these functions, e.g. as a ratio of Pfaffians, a sum over set-valued shifted tableaux, a sum over excited Young diagrams. As a geometric application, we show that these functions represent the Schubert classes in the $K$-theory of torus equivariant coherent sheaves on the maximal isotropic Grassmannians of symplectic and orthogonal types. This generalizes a corresponding result for the equivariant cohomology given by the authors. We also discuss a cancellation property enjoyed by these functions, which we call $K$-supersymmetric property. We prove that the $K$-theoretic $P$-functions form a (formal) basis of the ring of $K$-supersymmetric functions.

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