2020-06-29

## Permuted basement Macdonald E polynomials

The permuted-basement Macdonald polynomials generalize the
non-symmetric Macdonald polynomials, by introducing an additional parameter $\sigma \in \symS_n,$
the *basement*.
They were introduced in [Fer11] by J. Ferreira, as eigenpolynomials of certain operators.
Later in [Ale19a], a combinatorial model was introduced.

**Proposition (See [CMW18]).**

Let $\mu$ be a partition, of length at most $n.$ Then

\[ \macdonaldP_\mu(\xvec;q,t) = \sum_{\sigma \in \symS_n} \macdonaldE^{\sigma}_\mu(\xvec;q,t). \]**Conjecture (Olya Mandelshtam, 2019 (personal communication)).**

Some computations suggest that for any composition $\mu,$ we have

\[ R_\mu(q,t) \macdonaldP_\mu(\xvec;q,t) = \sum_{\sigma \in \symS_n} \macdonaldE^{\sigma}_\mu(\xvec;q,t) \]where $R_\mu(q,t)$ is in $\setQ(q,t).$

## Quasisymmetric Macdonald E polynomials

A quasisymmetric version of the non-symmetric Macdonald polynomials were introduced in [CHMMW19b].

They specialize to the quasisymmetric Schur polynomials at $q=t=0.$

**Conjecture (Alexandersson 2020).**

Let $\alpha$ be a composition. Then the coefficients $K_{\alpha\gamma}(q)$ in the expansion

\[ \macdonaldEQuasi_\alpha(\xvec;q,0) = \sum_{\gamma} K_{\alpha\gamma}(q) \schurQS_\gamma(\xvec) \]are in $\setN[q].$ Note that this resemblence the fact that $\macdonaldE_\alpha(\xvec;q,0)$ are key-positive, with versions of Kostka–Foulkes polynomials as coefficients.

## References

- [Ale19a] Per Alexandersson. Non-symmetric Macdonald polynomials and Demazure–Lusztig operators. Séminaire Lotharingien de Combinatoire, 76, 2019.
- [AS17] 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.
- [CHMMW19b] 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.
- [Fer11] Jeffrey Paul Ferreira. Row-strict quasisymmetric Schur functions, characterizations of Demazure atoms, and permuted basement nonsymmetric Macdonald polynomials. University of California Davis. 2011.