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 $\schurKQ_{\lambda}(\xvec).$

## References

- [BR12] Anders Skovsted Buch and Vijay Ravikumar. Pieri rules for the K-theory of cominuscule Grassmannians. Journal für die reine und angewandte Mathematik (Crelles Journal), 2012(668), January 2012.
- [CTY14] Edward Clifford, Hugh Thomas and Alexander Yong. K-theoretic Schubert calculus for $OG(n, 2n+1)$ and jeu de taquin for shifted increasing tableaux. Journal für die reine und angewandte Mathematik (Crelles Journal), 2014(690), January 2014.
- [HKPWZZ17] Zachary Hamaker, Adam Keilthy, Rebecca Patrias, Lillian Webster, Yinuo Zhang and Shuqi Zhou. Shifted Hecke insertion and the K-theory of $OG(n,2n+1)$. Journal of Combinatorial Theory, Series A, 151:207–240, October 2017.
- [IN13] Takeshi Ikeda and Hiroshi Naruse. K-theoretic analogues of factorial Schur P- and Q-functions. Advances in Mathematics, 243:22–66, August 2013.