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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 [AS17aAS19a].

A good overview and introduction to this topic, is [GR21aGR21b].

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 [HHL08Mas09]. 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.

$6$$5$$5$ 
$5$   
$4$   
$3$$3$$4$$2$
$2$$2$  
$1$$1$$6$ 
  $2$   
$6$ $4$  $5$
$1$$2$$3$  $5$
$1$$2$$3$$4$$5$$6$

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.

$a$
$\vdots$
$b$
   or   
$a$ 
$\vdots$ 
 $b$

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.

Type $A$:
$a_3$$b_1$
 $\vdots$
 $c_2$
   Type $B$:
$c_2$ 
$\vdots$ 
$a_3$$b_1$

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 $B$).

$\underline{4}$$\underline{2}$$1$$\textbf{2}$$4$
$5$$5$   
$3$$3$$\textbf{4}$$3$ 
$2$    
$1$$\underline{1}$   

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

\begin{equation} \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)}}. \end{equation}

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

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

see also [Eq. 5.7.8, Mac96a].

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 $q=t=0.$

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