Permuted basement Macdonald E polynomials

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#Permuted basement Macdonald E polynomials

The permuted-basement Macdonald polynomials (also called relative Macdonald polynomials [GR21]) 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 [Ale19], a combinatorial model was introduced. For some properties of these formulas, see [AS17, AS19].

A good overview and introduction to this topic is [GR21, GR21].

S. Corteel, O. Mandelshtam, and L. Williams connect multiline queues with Macdonald polynomials via the exclusion process [CMW18].

#Permuted-basement $t$-atoms

At $q=0,$ the permuted-basement Macdonald polynomials become permuted-basement $t$-atoms. Alexandersson and Sawhney study these polynomials using basement-permuting operators and fillings [AS19].

#Permuted-basement atoms

Setting $t=0$ in the permuted-basement $t$-atoms gives permuted-basement atoms. In the basement convention of [AS19], the identity basement gives ordinary Demazure atoms, while the longest basement gives key polynomials.

#Demazure $t$-atoms

The identity-basement case of the permuted-basement $t$-atoms is the family of Demazure $t$-atoms. They specialize at $t=0$ to ordinary Demazure atoms. These are not the same as the generalized Demazure atoms of J. Haglund, S. Mason, and J. B. Remmel.

#Definitions

The following section is mainly from [Ale19], with the filling product written in the Ferreira–Moura–Mandelshtam convention.

Let $\sigma = (\sigma_1,\dotsc,\sigma_n)$ be a list of $n$ distinct positive integers and let $\alpha=(\alpha_1,\dotsc,\alpha_n)$ 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 [HHL08, Mas09]. 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 at the bottom of the diagram, thus explaining the peculiar choice of terminology.

G. Z. D. e. Moura and O. Mandelshtam give probabilistic entry-swapping bijections for non-attacking fillings [MM25].

Definition

Let $F$ be an augmented filling. Two boxes $a$ and $b$ 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$

$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 \[\begin{aligned} \{ (r', c) \in \alpha : r \lt{} r' \text{ and } \alpha_{r'} \leq \alpha_r \} \text{ and } \\ \{ (r', c-1) \in \alpha : r' \lt{} r \text{ and } \alpha_{r'} \lt{} \alpha_r \}. \end{aligned}\]

The major index of an augmented filling $F$ is defined as \[\begin{aligned} \maj(F) = \sum_{ u \in \Des(F) } \leg(u)+1. \end{aligned}\] 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 that are not inversion triples.

Let $\mathrm{NAF}_\sigma(\alpha)$ denote the set of all non-attacking fillings of shape $\alpha,$ augmented with the basement $\sigma \in \symS_n,$ with all entries in the fillings 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

\[\macdonaldE^\sigma_\alpha(\xvec; q,t) = \sum_{ F \in \mathrm{NAF}_\sigma(\alpha)} \xvec^F q^{\maj(F)} t^{\coinv(F)} \prod_{ \substack{ u \in F \\ u \text{ is not in the basement} \\ F(d(u))\neq F(u) }} \frac{1-t}{1-q^{1+\leg(u)} t^{1+\arm(u)}}.\] The product is over all boxes $u$ in $F$ not in the basement, satisfying $F(d(u))\neq F(u).$

Remark (Source conventions).

The product range in the displayed formula above is the one used by Ferreira and by Moura–Mandelshtam [Def. 2.1, MM26]. The printed formula in [Def. 5, Ale19] includes basement boxes in this product. This should not be used literally: basement boxes may enter the arm sets of ordinary boxes, but they are not themselves product indices. This is also the convention implemented in Sage’s nonsymmetric Macdonald polynomial code [Sag19].

There is a second convention issue in the operator formula. In [Rem. 2.7, MM25], Moura–Mandelshtam note that the published definition of $\mathrm{twinv}(\alpha,\sigma)$ in [Ale19] has the inequality involving $\sigma$ reversed. The corrected convention is \[\mathrm{twinv}(\alpha,\sigma) \coloneqq \left|\{(i,j): i\lt{}j,\ \alpha_i\geq\alpha_j,\ \sigma_i\lt{}\sigma_j\}\right|.\]

Example (A four-filling check).

Let $\alpha=(1,1,0,1)$ and let $\sigma=(2,4,1,3).$ In the English convention used here, the four non-attacking fillings in $\mathrm{NAF}_\sigma(\alpha)$ are \[\begin{aligned} F_1&:\ (2\mid 1),\ (4\mid 4),\ (1\mid \emptyset),\ (3\mid 3),\\ F_2&:\ (2\mid 2),\ (4\mid 1),\ (1\mid \emptyset),\ (3\mid 3),\\ F_3&:\ (2\mid 2),\ (4\mid 4),\ (1\mid \emptyset),\ (3\mid 3),\\ F_4&:\ (2\mid 4),\ (4\mid 1),\ (1\mid \emptyset),\ (3\mid 3). \end{aligned}\] Their data, in the order displayed, are \[\begin{array}{c|c|c|c|c} & \xvec^F & \maj(F) & \coinv(F) & \text{ordinary product factors} \\ \hline F_1 & x_1x_3x_4 & 0 & 0 & \dfrac{1-t}{1-qt^3} \\ F_2 & x_1x_2x_3 & 0 & 1 & \dfrac{1-t}{1-qt^2} \\ F_3 & x_2x_3x_4 & 0 & 0 & 1 \\ F_4 & x_1x_3x_4 & 1 & 2 & \dfrac{1-t}{1-qt^3}\dfrac{1-t}{1-qt^2}. \end{array}\] Thus the filling formula gives \[\begin{aligned} \macdonaldE^{(2,4,1,3)}_{(1,1,0,1)}(\xvec;q,t) ={}& x_2x_3x_4 + x_1x_3x_4\frac{1-t}{1-qt^3} + x_1x_2x_3\,t\frac{1-t}{1-qt^2} \\ &\quad + x_1x_3x_4\,q t^2 \frac{(1-t)^2}{(1-qt^3)(1-qt^2)}. \end{aligned}\] This is the computation in [Ex. 2.13, MM25]. The third filling is the useful sanity check for the corrected product range: all ordinary entries equal the entry immediately to their left, so its product factor is $1.$

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].

A formula for the $\macdonaldE^\sigma_\alpha$ using set-valued tableaux can be found in [Thm. 2.2, DR22]. The proof relies on a bijection with alcove walks. O. Mandelshtam, H. Niergarth, and K. Singh study the $t=0$ specialization of these polynomials [MNS26]. They prove that every permuted basement Macdonald polynomial has a positive expansion in Demazure atoms at $t=0,$ and describe the structure coefficients using charge on a restricted set of semistandard tableaux. D. Orr and J. Rivera refine the Concha–Lapointe partially symmetric Macdonald identity to a subfamily of permuted basement Macdonald polynomials and give a combinatorial proof [OR25]. They also relate the Concha–Lapointe identity to invariance of normalized partially symmetric Macdonald polynomials under the Kazhdan–Lusztig involution.

#Properties

Proposition (See [CMW18]).

Let $\mu$ be a partition of length at most $n,$ and let $\macdonaldP_\mu(\xvec;q,t)$ denote the Macdonald P polynomials. Then \[\macdonaldP_\mu(\xvec;q,t) = \sum_{\sigma \in \symS_n(\mu)} \macdonaldE^{\rev(\sigma)}_{inc(\mu)}(\xvec;q,t),\] where $inc(\mu)$ is the list of entries of $\mu$ sorted in increasing order.

Conjecture (Olya Mandelshtam, 2019 personal communication).

For any fixed composition $\mu,$ we can find coefficients $R_{\mu}(q,t) \in \setQ(q,t)$ and some subset $M \subseteq \symS_n$ such that \[R_{\mu}(q,t) \macdonaldP_{\lambda(\mu)}(\xvec;q,t) = \sum_{\sigma \in M} \macdonaldE^{\sigma}_{\mu}(\xvec;q,t).\]

Combining [Prop. 1.1, GR21] with [Eq. 1.10, GR21], one obtains an expansion \[\macdonaldP_{\lambda(\mu)}(\xvec;q,t) = \sum_{\sigma \in \symS_n} R'_{\mu,\alpha}(q,t) \macdonaldE^{\sigma}_\mu(\xvec;q,t),\] see also [Eq. 5.7.8, Mac96].

#Shape changing identities

In [MM26, MM25], the authors extend the row-swapping identities for the $q=0$ specialization in [Prop. 5.5, AS19] to the full two-parameter family. Here $s_i\cdot\alpha$ is obtained from $\alpha$ by swapping $\alpha_i$ and $\alpha_{i+1},$ while $\sigma s_i$ is obtained from $\sigma$ by swapping $\sigma_i$ and $\sigma_{i+1}.$

Theorem ([Thm. 1.1, MM26]).

Let $\alpha$ be a weak composition with $n$ parts, and suppose that $\alpha_i=\alpha_{i+1}$ for some $i\in[n-1].$ Then, for every $\sigma\in\symS_n,$ \[\macdonaldE^\sigma_\alpha(\xvec;q,t) = \macdonaldE^{\sigma s_i}_\alpha(\xvec;q,t).\]

Theorem ([Thm. 1.3, MM26]).

Let $\alpha$ be a weak composition with $n$ parts, and suppose that $\alpha_i\gt{}\alpha_{i+1}$ for some $i\in[n-1].$ Then, for every $\sigma\in\symS_n,$ \[\macdonaldE^\sigma_\alpha(\xvec;q,t) = \macdonaldE^{\sigma s_i}_{s_i\cdot\alpha}(\xvec;q,t) + c_{i,\alpha,\sigma}(q,t) \macdonaldE^\sigma_{s_i\cdot\alpha}(\xvec;q,t),\] where \[c_{i,\alpha,\sigma}(q,t) \coloneqq t^{\mathrm{twinv}(s_i\cdot\alpha,\sigma)-\mathrm{twinv}(\alpha,\sigma)} \frac{1-t}{1-q^{\leg(u)+1}t^{\arm(u)}} \begin{cases} 1, & \text{if } \sigma_i\gt{}\sigma_{i+1},\\ q^{\leg(u)+1}t^{\arm(u)}, & \text{if } \sigma_i\lt{}\sigma_{i+1}, \end{cases}\] in which $u=(i+1,\alpha_{i+1}+1)$ is viewed as a box in the diagram $s_i\cdot\alpha,$ and $\leg(u)$ and $\arm(u)$ are computed in that diagram. Furthermore, \[\mathrm{twinv}(\alpha,\sigma) \coloneqq \left|\{(k,l): k\lt l,\ \alpha_k\geq \alpha_l,\ \sigma_k\lt\sigma_l\}\right|.\]

#Quasisymmetric Macdonald E polynomials

The quasisymmetric Macdonald E polynomials are a quasisymmetric version of the non-symmetric Macdonald polynomials introduced in [CHMM+19].

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 resembles the fact that $\macdonaldE_\alpha(\xvec;q,0)$ are key-positive, with versions of Kostka–Foulkes polynomials as coefficients.

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