The symmetric functions catalog

An overview of symmetric functions and related topics

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.

See [AS17AS19a].

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