2022-12-09

## Macdonald E polynomials

The non-symmetric Macdonald polynomials were introduced in [Opd95],
[Mac96a] and [Che95b].
The non-symmetric Macdonald polynomials are closely related with *affine root systems*,
and the *double affine Hecke algebra*. Their definition is rather indirect,
and does not give an efficient way of computing these non-symmetric polynomials.

In [HHL08], J. Haglund, M. Haiman and N. Loehr
found an explicit formula for the non-symmetric Macdonald polynomials in type $A,$
as a sum over *non-attacking fillings*.
This model is the basis for the permuted basement Macdonald polynomials,
and we refer to that page for definitions.

Later, an alternative combinatorial formula using alcove walks was
proved by A. Ram and M. Yip [RY11], which works for all Lie types.
At $q=t=0,$ their model reduces to the *Littelmann path model*.

In [BW19], an integrable vertex model is used to give an alternative formula for non-symmetric Macdonald polynomials.

See the book [Hag07] for more background on the type $A$ non-symmetric Macdonald polynomials. My personal research with non-symmetric Macdonald polynomials and further generalizations can be found in [Ale19aAS17aAS19a].

A model using multiline queues is introduced in [CMW18], and further explored in [CHMMW19a].

### Monk's rule

W. Baratta [Bar09Bar11] give Monk type rules (terminology from Schubert calculus) for products of the form

\[ x_j \macdonaldE_\mu(x_1,\dotsc,x_n;q,t), \quad \elementaryE_1(\xvec) \cdot \macdonaldE_\mu(x_1,\dotsc,x_n;q,t), \quad \elementaryE_{r}(\xvec) \cdot\macdonaldE_\mu(x_1,\dotsc,x_n;q,t), \]expanded again in the $\{\macdonaldE_\alpha\}$ basis. Baratta uses interpolation Macdonald polynomials in the proof.

In [HR22], the authors prove Monk type formulas for

\[ x_j \macdonaldE_\mu, \qquad (x_1+\dotsb + x_j) \macdonaldE_\mu, \qquad x_j^{-1} \macdonaldE_\mu, \qquad (x_j^{-1}+\dotsb + x^{-1}_j) \macdonaldE_\mu. \]Halverson–Ram uses a different method based on intertwiners.

## Non-symmetric q-Whittaker polynomials

The non-symmetric $q$-Whittaker polynomials are obtained by letting $t=0$ in $\macdonaldE_\mu(\xvec;q,t).$ The result can be though of as a non-symmetric analogue of the transformed Hall–Littlewood polynomials, and generalize the $q$-Whittaker functions.

The polynomials $\macdonaldE_\mu(\xvec;q,0)$ expand positively in key polynomials, and the coefficients are Kostka–Foulkes polynomials, see [AS17aAss18aAG18].

Models using *quantum alcove walks* and *quantum Lakshmibai–Seshadri path*
are considered in [LNSSS17].

## References

- [AG18] Sami Assaf and Nicolle S. González. Crystal graphs, key tabloids, and nonsymmetric Macdonald polynomials. 30th International Conference on Formal Power Series and Algebraic Combinatorics. Séminaire Lotharingien de Combinatoire, 80B(90) 2018. 12 pages
- [Ale19a] Per Alexandersson. Non-symmetric Macdonald polynomials and Demazure–Lusztig operators. Séminaire Lotharingien de Combinatoire, 76, 2019.
- [AS17a] Per Alexandersson and Mehtaab Sawhney. A major-index preserving map on fillings. Electronic Journal of Combinatorics, 24(4):1–30, 2017.
- [AS19a] Per Alexandersson and Mehtaab Sawhney. Properties of non-symmetric Macdonald polynomials at $q=1$ and $q=0$. Annals of Combinatorics, 23(2):219–239, May 2019.
- [Ass18a] Sami Assaf. Nonsymmetric Macdonald polynomials and a refinement of Kostka–Foulkes polynomials. Transactions of the American Mathematical Society, 370(12):8777–8796, July 2018.
- [Bar09] W. Baratta. Pieri-type formulas for nonsymmetric Macdonald polynomials. International Mathematics Research Notices, March 2009.
- [Bar11] W. Baratta. Further Pieri-type formulas for the nonsymmetric Macdonald polynomial. Journal of Algebraic Combinatorics, 36(1):45–66, October 2011.
- [BW19] Alexei Borodin and Michael Wheeler. Nonsymmetric Macdonald polynomials via integrable vertex models. arXiv e-prints, 2019.
- [Che95b] Ivan Cherednik. Nonsymmetric Macdonald polynomials. International Mathematics Research Notices, 1995(10):483, 1995.
- [CHMMW19a] Sylvie Corteel, James Haglund, Olya Mandelshtam, Sarah Mason and Lauren Williams. Compact formulas for Macdonald polynomials and quasisymmetric Macdonald polynomials. arXiv e-prints, 2019.
- [CMW18] Sylvie Corteel, Olya Mandelshtam and Lauren Williams. From multiline queues to Macdonald polynomials via the exclusion process. arXiv e-prints, 2018.
- [Hag07] James Haglund. The $q,t$-Catalan numbers and the space of diagonal harmonics (University lecture series). American Mathematical Society, 2007.
- [HHL08] James Haglund, Mark D. Haiman and Nicholas A. Loehr. A combinatorial formula for nonsymmetric Macdonald polynomials. American Journal of Mathematics, 130(2):359–383, 2008.
- [HR22] Tom Halverson and Arun Ram. Monk rules for type $GL_n$ Macdonald polynomials. arXiv e-prints, 2022.
- [LNSSS17] C. Lenart, S. Naito, D. Sagaki, A. Schilling and M. Shimozono. A uniform model for Kirillov–Reshetikhin crystals III: Nonsymmetric Macdonald polynomials at $t=0$ and Demazure characters. Transformation Groups, 22(4):1041–1079, February 2017.
- [Mac96a] Ian G. Macdonald. Affine Hecke algebras and orthogonal polynomials. In Séminaire Bourbaki. Société Mathématique de France, 1996.
- [Opd95] Eric M. Opdam. Harmonic analysis for certain representations of graded Hecke algebras. Acta Mathematica, 175(1):75–121, 1995.
- [RY11] Arun Ram and Martha Yip. A combinatorial formula for Macdonald polynomials. Advances in Mathematics, 226(1):309–331, 2011.