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

TL;DR

This paper introduces new symmetric functions called $K$-theoretic factorial Schur $P$- and $Q$-functions, providing combinatorial formulas and geometric interpretations in $K$-theory of isotropic Grassmannians, extending previous cohomology results.

Contribution

The authors define $K$-theoretic analogues of factorial Schur $P$- and $Q$-functions, establish their combinatorial expressions, and demonstrate their role as Schubert classes in $K$-theory of isotropic Grassmannians.

Findings

Derived combinatorial formulas as Pfaffian ratios and excited Young diagrams.

Proved these functions represent Schubert classes in $K$-theory.

Established the $K$-theoretic $Q$-cancellation property and basis results.

Abstract

We introduce two families of symmetric functions generalizing the factorial Schur $P$- and $Q$- functions due to Ivanov. We call them $K$-theoretic analogues of factorial Schur $P$- and $Q$- functions. We prove various combinatorial expressions for these functions, e.g. as a ratio of Pfaffians, and 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 remarkable property enjoyed by these functions, which we call the $K$-theoretic $Q$-cancellation property. We prove that the $K$-theoretic $P$-functions form a (formal) basis of the ring of functions with the $K$-theoretic $Q$-cancellation property.