Expanding K-theoretic Schur Q-functions

TL;DR

This paper develops new identities and formulas for $K$-theoretic Schur $P$- and $Q$-functions, including an expansion formula conjectured by Lewis and Naruse, advancing the understanding of their algebraic structure.

Contribution

It proves a conjectured expansion formula for $K$-theoretic Schur $Q$-functions in terms of $P$-functions and explores related identities and conjectures in the theory.

Findings

Derived identities involving $K$-theoretic Schur functions.

Proved a shifted skew Cauchy identity for symmetric Grothendieck polynomials.

Discussed conjectural formulas implying basis properties of $K$-theoretic Schur $Q$-functions.

Abstract

We derive several identities involving Ikeda and Naruse's $K$-theoretic Schur $P$- and $Q$-functions. Our main result is a formula conjectured by Lewis and the second author which expands each $K$-theoretic Schur $Q$-function in terms of $K$-theoretic Schur $P$-functions. This formula extends to some more general identities relating the skew and dual versions of both power series. We also prove a shifted version of Yeliussizov's skew Cauchy identity for symmetric Grothendieck polynomials. Finally, we discuss some conjectural formulas for the dual $K$-theoretic Schur $P$- and $Q$-functions of Nakagawa and Naruse. We show that one such formula would imply a basis property expected of the $K$-theoretic Schur $Q$-functions.