# The symmetric functions catalog

An overview of symmetric functions and related topics

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].