2020-02-05

## P-Grothendieck polynomials

The $P$-Grothendieck polynomials are symmetric, non-homogeneous symmetric functions, defined via shifted, set-valued tableaux. In [Haw20] the author introduce the weak $P$-Grothendieck polynomials, which plays the role of dual stable Grothendieck polynomials. The author show that these are Schur-P positive. This is analogous to the dual stable Grothendieck polynomials being Schur-positive.

Suppose $\mu$ is a strict partition with $m$ parts. The weak symmetric $P$-Grothendieck polynomial $\grothendieckPDual_\mu(\xvec)$ in $n \geq m$ variables is defined as

\[ \grothendieckPDual_\mu(\xvec) = \prod_{i\lt j} (x_i-x_j)^{-1} \sum_{\sigma \in \symS_n / \symS_{n-m}} \sign(\sigma) \left( \prod_i \left( \frac{x_{\sigma_i}}{1-x_{\sigma_i}} \right)^{\mu_i} \right) \left( \prod_{i \lt j, i \leq m} x_{\sigma_i} + x_{\sigma_j} \right) \left( \prod_{m \lt i \lt j} x_{\sigma_i} - x_{\sigma_j} \right). \]Here, $\symS_n / \symS_{n-m}$ is the set of permutations in $\symS_n$ with no descent after position $m.$

## References

- [Haw20] Graham Hawkes. $p$-Schur positive $p$-Grothendieck polynomials. arXiv e-prints, 2020.