# The symmetric functions catalog

An overview of symmetric functions and related topics

2019-06-06

## K-theoretic Schur P/Q polynomials

In [IN13], Ikeda and Naruse introduce $K$-theoretic analogues of the Schur $P$ and Schur $Q$ functions, as well as factorial versions of these. The factorial versions are denoted $\schurKP_{\lambda}(\xvec|b)$ and $\schurKQ_{\lambda}(\xvec|b),$ where $b = (b_1,b_2,\dotsc)$ is a vector of parameters. When all these are set to $0,$ we obtain $\schurKP_{\lambda}(\xvec)$ and $\schurKQ_{\lambda}(\xvec).$ These are $K$-theoretic analogues of $\schurP_{\lambda}(\xvec)$ and $\schurQ_{\lambda}(\xvec).$

They give a presentation for equivariant $K$-theory classes of the orthogonal and Lagrangian Grassmannian.

### Properties

The fundamental quasisymmetric expansion of $\schurKP_{\lambda}(\xvec)$ and $\schurKQ_{\lambda}(\xvec)$ can be found in [HKPWZZ17].

### Littlewood–Richardson rule

The Littlewood-Richardson rule for $\schurKP_{\lambda}(\xvec)$ is due to Clifford-Thomas-Yong [CTY14]. Their result is based on a Pieri rule due to Buch and Ravikumar, [BR12].

Problem (Littlewood–Richardson rule).

Find a Littlewood–Richardson rule for $\schurKQ_{\lambda}(\xvec).$