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2022-03-04

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 [Fer11b] by J. Ferreira, as eigenpolynomials of certain operators. Later in [Ale19a], a combinatorial model was introduced. For some properties of these formulas, see .

A good overview and introduction to this topic, is .

Definitions

The following section is mainly from [Ale19a].

Let $\sigma = (\sigma_1,\dots,\sigma_n)$ be a list of $n$ different positive integers and let $\alpha=(\alpha_1,\dots,\alpha_n)$ be a weak integer composition. An augmented filling of shape $\alpha$ and basement $\sigma$ is a filling of a Young diagram of shape $(\alpha_1,\dotsc,\alpha_n)$ with positive integers, augmented with a zeroth column filled from top to bottom with $\sigma_1,\dotsc,\sigma_n.$

Note that we use English notation rather than the skyline fillings used in . For example, the following figure illustrates the difference, where the English notation is used in the left diagram, while the skyline convention is used in the right diagram.

In the skyline convention, the basement appears in the bottom of the diagram, thus explaining the peculiar choice of terminology.

Definition.

Let $F$ be an augmented filling. Two boxes $a,$ $b,$ are attacking if $F(a)=F(b)$ and the boxes are either in the same column, or they are in adjacent columns, with the rightmost box in a row strictly below the other box.

A filling is non-attacking if there are no attacking pairs of boxes.

Definition.

A triple of type $A$ is an arrangement of boxes, $a,$ $b,$ $c,$ located such that $a$ is immediately to the left of $b,$ and $c$ is somewhere below $b,$ and the row containing $a$ and $b$ is at least as long as the row containing $c.$ Similarly, a triple of type $B$ is an arrangement of boxes, $a,$ $b,$ $c,$ located such that $a$ is immediately to the left of $b,$ and $c$ is somewhere above $a,$ and the row containing $a$ and $b$ is strictly longer than the row containing $c.$

A type $A$ triple is an inversion triple if the entries ordered increasingly form a counter-clockwise orientation. Similarly, a type $B$ triple is an inversion triple if the entries ordered increasingly form a clockwise orientation. If two entries are equal, the one with largest subscript in the figures below is considered largest.

If $u = (i,j)$ let $d(u)$ denote $(i,j-1).$ A descent in $F$ is a non-basement box $u$ such that $F(d(u)) \lt F(u).$ The set of descents in $F$ is denoted $\Des(F).$

Example.

Here is a non-attacking filling of shape $(4,1,3,0,1)$ and basement $(4,5,3,2,1).$ The bold entries are descents and the underlined entries form a type $A$ inversion triple. There are in total $7$ inversion triples (of type $A$ and $B$).

The leg, $\leg(u),$ of a box $u$ in a diagram is the number of boxes to the right of $u$ in the diagram. The arm, denoted $\arm(u),$ of a box $u = (r,c)$ in a diagram $\alpha$ is defined as the cardinality of the sets

\begin{align*} \{ (r', c) \in \alpha : r < r' \text{ and } \alpha_{r'} \leq \alpha_r \} \text{ and } \\ \{ (r', c-1) \in \alpha : r' < r \text{ and } \alpha_{r'} < \alpha_r \}. \end{align*}

The major index of an augmented filling $F$ is defined as

\begin{align*} \maj(F) = \sum_{ u \in \Des(F) } \leg(u)+1. \end{align*}

The number of inversions, $\inv(F)$ of a filling is the number of inversion triples of either type. The number of coinversions, $\coinv(F),$ is the number of type $A$ and type $B$ triples which are not inversion triples.

Let $\mathrm{NAF}_\sigma(\alpha)$ denote all non-attacking fillings of shape $\alpha,$ augmented with the basement $\sigma \in \symS_n,$ and all entries in the fillings are from $[n].$

Definition.

Let $\sigma \in \symS_n$ and let $\alpha$ be a weak composition with $n$ parts. The non-symmetric permuted basement Macdonald polynomial $\macdonaldE^\sigma_\alpha(\xvec;q,t)$ is defined as

$$\macdonaldE^\sigma_\alpha(\xvec; q,t) = \sum_{ F \in \mathrm{NAF}_\sigma(\alpha)} \xvec^F q^{\maj(F)} t^{\coinv(F)} \prod_{ \substack{ u \in F \\ u \text{ is in the basement or} \\ F(d(u))\neq F(u) }} \frac{1-t}{1-q^{1+\leg(u)} t^{1+\arm(u)}}.$$

The product is over all boxes $u$ in $F$ such that either $u$ is in the basement or $F(d(u))\neq F(u).$

When $\sigma = \omega_0,$ we recover the non-symmetric Macdonald polynomials defined in [HHL08], $\macdonaldE_\alpha(\xvec;q,t).$ There is a slight difference in notation, the index $\alpha$ is reversed compared to [HHL08].

Properties

Proposition (See [CMW18]).

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

$\macdonaldP_\lambda(\xvec;q,t) = \sum_{\mu\in \symS_n(\lambda)} \macdonaldE^{\sigma}_{inc(\mu)}(\xvec;q,t),$

where $\sigma$ is the longest permutation which sorts the entries of $\mu$ in increasing order, i.e.,

$\mu_{\sigma(1)} \leq \mu_{\sigma(2)} \leq \dotsb \leq \mu_{\sigma(n)}.$
Conjecture (Olya Mandelshtam, 2019 personal communication).

We can find coefficients $R_\mu(q,t) \in \setQ(q,t),$ such that for any composition composition $\mu,$ we have

$R_\mu(q,t) \macdonaldP_\mu(\xvec;q,t) = \sum_{\mu\in \symS_n(\lambda)} \macdonaldE^{\sigma}_{inc(\mu)}(\xvec;q,t).$

Combining [Prop 1.1, GR21b] with [Eq 1.10, GR21b], one can more or less find the expansion

$\macdonaldP_\mu(\xvec;q,t) = \sum_{\sigma \in \symS_n} R'_{\mu,\alpha}(q,t) \macdonaldE^{\sigma}_\mu(\xvec;q,t),$

Quasisymmetric Macdonald E polynomials

A quasisymmetric version of the non-symmetric Macdonald polynomials were introduced in [CHMMW19a].

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

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.