2021-10-04

## Lascoux polynomials

The Lascoux polynomials, is a family of non-symmetric, non-homogeneous polynomials, first introduced by A. Lascoux in [Las04]. The set $\{ \lascoux_\alpha(x_1,\dotsc,x_n) \}_\alpha$ where $\alpha$ ranges over all compositions of length $n,$ is a basis for $\setC[x_1,\dotsc,x_n].$

The Lascoux polynomials are the K-theoretical analog of the key polynomials, and they generalize the Grothendieck polynomials. Similarly, there are Lascoux-atom polynomials, which are K-theoretical analogs of the Demazure atom polynomials.

*Note:* There seem to be other types of polynomials referred to as Lascoux polynomials,
e.g. [BCMVV21].

### Definition

We use the same notation as in the operator definition of key polynomials. We introduce the $\beta$-versions of the divided difference operator $\partial_i,$ and $\pi_i$:

\[ \partial_i^{(\beta)}(f) \coloneqq \partial_i(f + \beta x_{i+1} f) \qquad \pi_i^{(\beta)}(f) \coloneqq \partial_i^{(\beta)}(x_i f). \]We then define the Lascoux polynomial as

\[ \lascoux^{(\beta)}_{\alpha}(\xvec) \coloneqq \begin{cases} \xvec^{\alpha} & \text{ if $\alpha$ is a partition} \\ \pi_i^{(\beta)} \lascoux^{(\beta)}_{s_i \alpha}(\xvec) & \text{ if $\alpha_i \lt a_{i+1}$}. \end{cases} \]Note that $\key_{\alpha}(\xvec) = \lascoux^{(0)}_{\alpha}(\xvec),$ that is, at $\beta=0$ we recover a key polynomial.

The first combinatorial model for Lascoux polynomials is given in [Thm. 4.1, BSW20],
where the Lascoux polynomials are presented as a sum
over set-valued tableaux. They also give a combinatorial formula involving
set-valued *skyline fillings* (terminology from non-symmetric Macdonald polynomials).
Another set-valued tableau formula is proved in [Thm. 1.1, RY21].

A set-valued tableau formula is also given in [Thm. 3.17, Yu21]. Here, the tableaux have composition shape, and the author argues that his formula is simpler, as it does not use the Lusztig involution.

In [MPS18], the question is raised if there is some type of K-theoretical crystal structure for Lascoux polynomials. This seems to be answered in [Yu21].

### Properties of Lascoux polynomials

The Schubert polynomials expand positively into key polynomials. In the same manner, in [SY21], M. Shimozono and T. Yu give a formula for the expansion of Grothendieck polynomials into Lascoux polynomials, thus proving an earlier conjecture by V. Reiner and A. Yong [RY21].

## Lascoux atom

### A vertex model for Lascoux atoms

In [BSW20], the authors construct a 5-vertex model whose partition function is the Lascoux-atoms. This is the first combinatorial model for the Lascoux-atoms.

## References

- [BCMVV21] Alessio Borzí, Xiangying Chen, Harshit J. Motwani, Lorenzo Venturello and Martin Vodička. The leading coefficient of Lascoux polynomials. arXiv e-prints, 2021.
- [BSW20] Valentin Buciumas, Travis Scrimshaw and Katherine Weber. Colored five-vertex models and Lascoux polynomials and atoms. Journal of the London Mathematical Society, 102(3):1047–1066, May 2020.
- [Las04] Alain Lascoux. Schubert & Grothendieck: Un bilan bidécennal. Séminaire Lotharingien de Combinatoire, 50:1–32, 2004.
- [MPS18] Cara Monical, Oliver Pechenik and Travis Scrimshaw. Crystal structures for symmetric Grothendieck polynomials. arXiv e-prints, 2018.
- [RY21] Victor Reiner and Alexander Yong. The "Grothendieck to Lascoux" conjecture. arXiv e-prints, 2021.
- [SY21] Mark Shimozono and Tianyi Yu. Grothendieck to Lascoux expansions. arXiv e-prints, 2021.
- [Yu21] Tianyi Yu. Set-valued tableaux rule for Lascoux polynomials. arXiv e-prints, 2021.