2022-07-20

## Monomial slide polynomials

The monomial slide polynomials were introduced by S. Assaf and D. Searles in [AS17b]. The slide polynomials form a basis for the space of polynomials, and can be seen as a lift of the monomial quasisymmetric functions.

**Definition.**

For $\alpha$ being a weak composition, we set

\[ \slideM_\alpha(\xvec) \coloneqq \sum_{\substack{b \trianglerighteq a \\ \mathrm{flat}(b)=\mathrm{flat}(a)}} \xvec^b \]where $\mathrm{flat}(\beta)$ denotes the composition obtained by removing all 0s from $\beta,$ and $\trianglerighteq$ denotes dominance order.

As an example (from [Eq. 3.7, AS17b]), $\slideM_{(0,2,0,3)}(\xvec) = x_1^2x_2^3 + x_1^2x_3^3 + x_1^2 x_4^3 + x_2^2 x_3^3 + x_2^2 x_4^3.$

The monomial slide polynomials have the monomial quasisymmetric functions as stable limit:

\[ \lim_{m \to \infty} \slideM_{0^m\times \alpha}(\xvec) = \qmonom_{\mathrm{flat}(\alpha)}(\xvec). \]## Fundamental slide polynomials

The *fundamental slide polynomials* (or sometimes just slide polynomials)
were introduced by S. Assaf and D. Searles in [AS17b].
The slide polynomials form a basis for the space of polynomials, and can be seen as a lift
of the gessel quasisymmetric functions.

The K-theoretical analog of fundamental slide polynomials are the \hyperref[glide]{glide polynomials}, see [PS17].

The fundamental slide polynomials expand positively in the monomial slide polynomials. Moreover, the Schubert polynomials expand positively in the fundamental slide polynomials, [thm. 3.13, AS17b]. The main motivation for introducing the fundamental slide polynomials, is that products of Schubert polynomials can be expanded (with a combinatorial formula) into fundamental slide polynomials.

**Definition.**

For $\alpha$ being a weak composition, fundamental slide polynomial $\slideF_\alpha$ is defined as

\[ \slideF_\alpha(\xvec) \coloneqq \sum_{\substack{b \trianglerighteq a \\ \mathrm{flat}(b)\text{ refines }\mathrm{flat}(a)}} \xvec^b \]where $\mathrm{flat}(\beta)$ denotes the composition obtained by removing all 0s from $\beta,$ and $\trianglerighteq$ denotes dominance order.

The fundamental slide polynomials have the gessel quasisymmetric functions as stable limit:

\[ \lim_{m \to \infty} \slideF_{0^m\times \alpha}(\xvec) = \gessel_{\mathrm{flat}(\alpha)}(\xvec). \]### Slide positive families

Key polynomials expand positively in the fundamental slide basis, [Thm. 2.13, AS18c]. In [CW22], the authors determine for which $\alpha,$ the key polynomials $\key_\alpha$ expanded into slide polynomials are multiplicity free.

In [AB19], the authors consider a flagged version of $(P,w)$-partitions, and show that these are slide-positive. In [TWZ22], it is shown that certain polynomials similar to chromatic symmetric functions are slide-positive.

See [ST21a] the notion of *slide complexes*.

## Lock polynomials

Lock polynomials were introduced by S. Assaf and D. Searles in [AS19c]. The Lock polynomials form a basis for the polynomial ring, and are indexed by weak compositions. The combinatorial formula for these is

\[ \lock_\alpha(\xvec) \coloneqq \sum_{T \in LT(\alpha)} \xvec^T \]where the sum is over all *lock tableaux*.

As an example (from [Wan20a]), we have

\[ \lock_{(0,2,3)}(\xvec) = x_2^2 x_3^2 + x_1x_2x_3^3 + x_1^2x_3^3 + x_1x_2^2x_3^2 + x_1^2x_2x_3^2 + x_1^2x_2^2x_3 + x_1^2 x_2^3 \]Whenever the non-zero parts of $\alpha$ are weakly decreasing, we have that the lock polynomial coincides with a key polynomial; $\lock_\alpha(\xvec)=\key_\alpha(\xvec),$ see [Thm. 6.12, AS19c].

There is a crystal structure on lock polynomials, explored by G. Wang in [Wan20a]. This crystal structure embeds naturally into Demazure crystals.

## Kohnert polynomials

Kohnert polynomials were introduced in [AS19c]. This is a large family of polynomials, which contains several of the classical families as special cases, in particular the family of Schubert polynomials.

All Kohnert polynomials are monomial slide positive, [Thm. 3.7, AS19c].

## References

- [AB19] Sami Assaf and Nantel Bergeron. Flagged $(\mathcal{ p },ρ)$-partitions. arXiv e-prints, 2019.
- [AS17b] Sami Assaf and Dominic Searles. Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams. Advances in Mathematics, 306:89–122, January 2017.
- [AS18c] Sami Assaf and Dominic Searles. Kohnert tableaux and a lifting of quasi-Schur functions. Journal of Combinatorial Theory, Series A, 156:85–118, May 2018.
- [AS19c] Sami Assaf and Dominic Searles. Kohnert polynomials. Experimental Mathematics, :1–27, April 2019.
- [CW22] Soojin Cho and Stephanie Van Willigenburg. Slide multiplicity free key polynomials. The Electronic Journal of Combinatorics, 29(1), January 2022.
- [PS17] Oliver Pechenik and Dominic Searles. Decompositions of Grothendieck polynomials. International Mathematics Research Notices, 2019(10):3214–3241, September 2017.
- [ST21a] E. Yu. Smirnov and A. A. Tutubalina. Slide polynomials and subword complexes. Sbornik: Mathematics, 212(10):1471–1490, October 2021.
- [TWZ22] Vasu Tewari, Andrew Wilson and Philip Zhang. Chromatic nonsymmetric polynomials of Dyck graphs are slide-positive. Proceedings of the American Mathematical Society, March 2022.
- [Wan20a] George Wang. Locks fit into keys: a crystal analysis of Lock polynomials. Annals of Combinatorics, 24(4):767–789, October 2020.