Combinatorial proof of a permuted basement Macdonald polynomial identity

TL;DR

This paper provides a combinatorial proof of a refined identity for permuted basement Macdonald polynomials, extending known symmetry properties and linking them to Kazhdan-Lusztig involution.

Contribution

It refines the Concha-Lapointe identity for permuted basement Macdonald polynomials and proves it combinatorially, connecting it to Kazhdan-Lusztig involution.

Findings

Refined the Concha-Lapointe identity for a sub-family of permuted basement Macdonald polynomials.

Provided a combinatorial proof of the refined identity.

Showed the identity's equivalence to invariance under Kazhdan-Lusztig involution.

Abstract

A well-known and fundamental property of the Macdonald polynomials $P_\lambda(x;q,t)$ is their invariance under the transformation sending $(q,t)$ to $(q^{-1},t^{-1})$. Recently, Concha and Lapointe showed that this property extends in an interesting, nontrivial way to an identity for partially symmetric Macdonald polynomials. Their identity played a key role in the work of Bechtloff Weising and Orr linking partially symmetric Macdonald polynomials to parabolic flag Hilbert schemes. In this paper, we refine the Concha-Lapointe identity to a sub-family of Alexandersson's permuted basement Macdonald polynomials and give a combinatorial proof of the refined identity. We show also that the Concha-Lapointe identity is equivalent to the assertion that (normalized) partially symmetric Macdonald polynomials are fixed under the Kazhdan-Lusztig involution.

Equations

$$P_{\lambda |\gamma }(x_1,\dots ,x_m,q{x}_n,\dots ,q{x}_{m+1};q^{-1},t^{-1})=q^{{\gamma }_1+\cdots +{\gamma }_{n-m}}{t}^{\mathrm{inv}(\gamma )-\ell ({\omega }_0^{[m+1,n]})}{T}_{{\omega }_0^{[m+1,n]}}{P}_{\lambda |\gamma }$$

$$P_{\lambda |\gamma }^{\ast }=t^{\mathrm{inv}(\lambda |\gamma )}{P}_{\lambda |\gamma }$$

$$T_{i}f=t{\sigma }_i(f)+(t-1)x_{i+1}\frac{{\sigma }_i(f)-f}{x_i-x_{i+1}}.$$

$$T_{\pi }=T_{i_1}\cdots T_{i_{\ell }}$$

$$P_{\lambda }=\sum _{\pi \in S_n^{\lambda }}{T}_{\pi }{E}_{\lambda }$$

$$T_i{E}_{\mu }=t{E}_{\mu }.$$

$$P_{\lambda |\gamma }=\sum _{\pi \in S_m^{\lambda }}{T}_{\pi }{E}_{\mu }$$

$$E_{\mu }^{\pi }=t^{-{\ell }_{\mu }(\pi )}{T}_{\pi }{E}_{\mu }$$

$${\mathrm{dg}}^{\prime }(\mu )=\{(i,j)\in (Z_{\ge 0}{)}^2:1\le i\le n,1\le j\le {\mu }_i\}$$

$$\widehat{\mathrm{dg}}(\mu )={\mathrm{dg}}^{\prime }(\mu )\cup \{(i,0):1\le i\le n\},$$

$$\text{leg}(u)$$

$${\text{arm}}^{\text{left}}(u)$$

$${\text{arm}}^{\text{right}}(u)$$

$$\text{arm}(u)$$

$$l(u)$$

$$a(u)$$

$$\sigma :{\mathrm{dg}}^{\prime }(\mu )\to [n],$$

$$\mathrm{Des}(\hat{\sigma })$$

$$\mathrm{Asc}(\hat{\sigma })$$

$$\mathrm{maj}(\hat{\sigma })$$

$$\mathrm{inv}(\hat{\sigma })$$

$$\mathrm{coinv}(\hat{\sigma })$$

$$E_{\mu }^{\pi }(x;q,t)=\sum _{\sigma \in {\mathrm{NAF}}_{\mu }^{\pi }}{x}^{\sigma }{q}^{\mathrm{maj}(\sigma )}{t}^{\mathrm{coinv}(\sigma )}\prod _{\begin{array}{c}u\in {\mathrm{dg}}^{\prime }(\mu )\\ \hat{\sigma }(u)\ne \hat{\sigma }(d(u))\end{array}}\frac{1-t}{1-q^{l(u)+1}{t}^{a(u)+1}},$$

$$w=d(u)\text{ and }v\in \text{arm}(u).$$

$${\chi }_{xy}(\hat{\sigma })=\{\begin{array}{ll}1& \text{if }\hat{\sigma }(x)>\hat{\sigma }(y)\\ 0& \text{otherwise}.\end{array}$$

$${\chi }_{uv}(\hat{\sigma })+{\chi }_{vw}(\hat{\sigma })-{\chi }_{uw}(\hat{\sigma })\in \{0,1\}.$$

$$\sum _{(u,v,w)}({\chi }_{uv}(\hat{\sigma })+{\chi }_{vw}(\hat{\sigma })-{\chi }_{uw}(\hat{\sigma })).$$

$${\mathrm{maj}}^{\prime }(\hat{\sigma })$$

$${\mathrm{coinv}}^{\prime }(\hat{\sigma })$$

$$E_{\mu }(x;q^{-1},t^{-1})=\sum _{\sigma \in {\mathrm{NAF}}_{\mu }^{\pi }}{x}^{\sigma }{q}^{{\mathrm{maj}}^{\prime }(\sigma )}{t}^{{\mathrm{coinv}}^{\prime }(\sigma )}\prod _{\begin{array}{c}u\in {\mathrm{dg}}^{\prime }(\mu )\\ \hat{\sigma }(u)\ne \hat{\sigma }(d(u))\end{array}}\frac{1-t}{1-q^{l(u)+1}{t}^{a(u)+1}}.$$

$${\pi }^c=(b+a-\pi (1))(b+a-\pi (2))\cdots (b+a-\pi (n)).$$

$$E_{\mu }^{{\pi }_1^c{\pi }_2^c}(x_1,\dots ,x_n;q,t)=q^{-p}{E}_{\mu }^{{\pi }_1{\pi }_2}(x_m,\dots ,x_1,q{x}_n,\dots ,q{x}_{m+1};q^{-1},t^{-1}).$$

$$|{\mathrm{NAF}}_{\mu }^{{\pi }_1{\pi }_2}|=|{\mathrm{NAF}}_{\mu }^{{\pi }_1^c{\pi }_2^c}|$$

$$f(\hat{\sigma }(u))=\{\begin{array}{ll}{\pi }_1^c\circ {\pi }_1^{-1}\circ \hat{\sigma }(u)& \text{if }\hat{\sigma }(u)\le m\\ {\pi }_2^c\circ {\pi }_2^{-1}\circ \hat{\sigma }(u)& \text{if }\hat{\sigma }(u)\ge m+1\end{array}$$

Full text

1 Introduction

The Macdonald polynomials $P_{\lambda}(x;q,t)$ [14]—which are symmetric polynomials in variables $x=(x_{1},\dotsc,x_{n})$ depending rationally on parameters $q,t$—satisfy a number of beautiful identities. One of these is their invariance under inversion of the parameters: $P_{\lambda}(x;q^{-1},t^{-1})=P_{\lambda}(x;q,t)$.

In this paper, we consider a larger family of Macdonald polynomials obtained by relaxing the requirement of symmmetry in the variables $x_{1},\dotsc,x_{n}$. Namely, for arbitrary $m$ with $0\leq m\leq n$, we consider polynomials which are symmetric in $x_{1},\dotsc,x_{m}$, with no condition of symmetry imposed upon $x_{m+1},\dotsc,x_{n}$. These are the partially symmetric Macdonald polynomials $P_{\lambda|\gamma}(x;q,t)$ studied in [7, 13], where $\lambda$ is a partition (having at most $m$ parts) and $\gamma$ is a composition (of length $n-m$). The $P_{\lambda|\gamma}$ interpolate between the symmetric $P_{\lambda}$ (when $m=n$) and the nonsymmetric Macdonald polynomials $E_{\gamma}$ [4] (when $m=0$).

One motivation for studying the partially symmetric polynomials $P_{\lambda|\gamma}$ is that many features of the $P_{\lambda}$ admit nontrivial extensions to the partially symmetric setting. For instance, Lapointe [13] formulated a positivity conjecture for the $P_{\lambda|\gamma}$ extending the well-known Macdonald positivity theorem proved by Haiman [10]. Subsequent work of Concha-Lapointe [6] established a number of fundamental properties of the $P_{\lambda|\gamma}$, including the following extension of the identity $P_{\lambda}(x;q,t)=P_{\lambda}(x;q^{-1},t^{-1})$:

Theorem 1.1** ([6]).**

For any partition $\lambda$ with at most $m$ parts and $\gamma=(\gamma_{1},\dotsc,\gamma_{n-m})\in(\operatorname{\mathbb{Z}}_{\geq 0})^{n-m}$, we have

[TABLE]

where $\omega_{0}^{[m+1,n]}\in S_{[m+1,n]}$ is the long element of the symmetric group of the set $[m+1,n]=\{m+1,m+2,\dotsc,n\}$, $T_{\omega_{0}^{[m+1,n]}}$ is the corresponding Demazure-Lusztig operator (see §2.3), and $\mathrm{inv}(\gamma)=|\{i<j:\gamma_{i}>\gamma_{j}\}|$.

We observe that equation (1.1) reduces to the aforementioned identity when $m=n$, since in that case $\gamma$ is empty. Moreover, when $m=0$, it produces a known identity for the nonsymmetric Macdonald polynomials; see [5, Prop. 3.3.3] or [1, Prop. 6] for the nonsymmetric identity and also [11, Theorem 4.8] for its connection to the Kazhdan-Lusztig involution. The shift of variables $x_{m+1},\dotsc,x_{n}$ by the parameter $q$ is an interesting feature of the partially symmetric setting. This shift is not present in the symmetric case, while in the nonsymmetric case it cancels with $q^{\gamma_{1}+\dotsm+\gamma_{m-n}}$ by homogeneity.

The general case of (1.1) played a key role in the work [15] of Bechtloff Weising and the first named author, where it was used to show that an explicit modified form of the $P_{\lambda|\mu}$ is sent, under a canonical isomorphism due Carlsson-Gorsky-Mellit [3], to the basis of torus fixed points for the equivariant $K$-theory of parabolic flag Hilbert schemes. (The precise statement proved in [15] was formulated as a conjecture by Goodberry and the first named author in [8].) In this context, the analog of Macdonald positivity [10] remains open; see [12, 13] for positivity conjectures and [2] for very recent progress involving newly introduced nonsymmetric LLT polynomials.

The proof of equation (1.1) in [6] is algebraic and employs a characterization of the $P_{\lambda|\gamma}$ as eigenfunctions of certain difference operators. Our original aim in this work was to give a combinatorial proof of (1.1) based on the Haglund-Haiman-Loehr formula [9], thereby explaining in combinatorial terms the peculiarities of (1.1) required by the partially symmetric setting. In the course of working out the details, we found that most natural statement involves an equality of summands from each side of (1.1) which are instances of Alexandersson’s permuted basement polynomials [1]. Our main result, Theorem 3.9 below, thus gives a refinement of (1.1) to these summands.

We also give the following interpretation of (1.1) in terms of the Kazhdan-Lusztig involution $*$, extending [11, Theorem 4.8] and providing yet another proof of the Concha-Lapointe identity:

Theorem 1.2**.**

Under the Kazhdan-Lusztig involution $*$ on $\mathbb{Q}(q,t)[x_{1},\dotsc,x_{n}]^{S_{m}}$ (see (5.5)),

[TABLE]

for all $\lambda|\gamma$ as in Theorem 1.1 above, where $\mathrm{inv}(\lambda|\gamma)=\mathrm{inv}(\gamma)+|\{(i,j)\in[m]\times[n-m]:\lambda_{i}>\gamma_{j}\}|$. Furthermore, equations (1.1) and (1.2) are equivalent.

Acknowledgements

We thank Milo Bechtloff Weising for related collaborations and discussions. D.O. gratefully acknowledges support from the Simons Foundation.

2 Macdonald polynomials

2.1 Notation and conventions

For all $a,b\in\mathbb{N}$ with $a\leq b$, let $[a,b]=\{a,a+1,\dotsc,b\}$ and $[b]=[1,b]=\{1,\dotsc,b\}$.

For any set $A$, let $S_{A}$ be the symmetric group of permutations of $A$, with the group operation given by composition: $\pi\rho=\pi\circ\rho$.

For an arbitrary $n\in\mathbb{N}$, let $\mathbb{Q}(q,t)[x_{1},\dotsc,x_{n}]$ be the algebra of polynomials in $x_{1},\dotsc,x_{n}$ over the field $\mathbb{Q}(q,t)$ of rational functions in indeterminates $q$ and $t$. The symmetric group $S_{n}=S_{[n]}$ acts naturally from the left on $\mathbb{Q}(q,t)[x_{1},\dotsc,x_{n}]$ by algebra automorphisms, which are specified by $\pi(x_{i})=x_{\pi(i)}$ for all $\pi\in S_{n}$ and $i\in[n]$. For any subgroup $G\subset S_{n}$, let $\mathbb{Q}(q,t)[x_{1},\dotsc,x_{n}]^{G}$ the subalgebra of polynomials which are invariant under $G$.

The group $S_{n}$ also acts naturally from the left on $\mathbb{Z}^{n}$, by the linear automorphisms $\pi(\mu)=(\mu_{\pi^{-1}(1)},\dotsc,\mu_{\pi^{-1}(n)})$ for all $\pi\in S_{n}$ and all $\mu=(\mu_{1},\dotsc,\mu_{n})\in\mathbb{Z}^{n}$.

2.2 Symmetric and nonsymmetric Macdonald polynomials

For an element $\lambda\in(\mathbb{Z}_{\geq 0})^{n}$ such that $\lambda_{1}\geq\dotsm\geq\lambda_{n}$, i.e., a partition with at most $n$ parts, denote by $P_{\lambda}=P_{\lambda}(x_{1},\dotsc,x_{n};q,t)$ the (symmetric) Macdonald polynomial of [14]. These form a $\mathbb{Q}(q,t)$-basis of the space $\mathbb{Q}(q,t)[x_{1},\dotsc,x_{n}]^{S_{n}}$ of symmetric polynomials.

For a composition $\mu\in(\mathbb{Z}_{\geq 0})^{n}$, denote by $E_{\mu}=E_{\mu}(x_{1},\dotsc,x_{n};q,t)$ the (type $GL_{n}$) nonsymmetric Macdonald polynomial as defined in [9]. These form a $\mathbb{Q}(q,t)$-basis of the space $\mathbb{Q}(q,t)[x_{1},\dotsc,x_{n}]$ of all polynomials.

2.3 Demazure-Lusztig operators

For $i=1,\dotsc,n-1$, let $\sigma_{i}=(i,i+1)\in S_{n}$ be the simple transposition and let $T_{i}$ be the Demazure-Lusztig operator on $\mathbb{Q}(q,t)[x_{1},\dotsc,x_{n}]$ given by

[TABLE]

For $\pi\in S_{n}$ and any reduced expression $\pi=\sigma_{i_{1}}\dotsc\sigma_{i_{\ell}}$, where $\ell=\ell(\pi)$ is the length of $\pi$, it is well-known that the operator $T_{\pi}$ defined by

[TABLE]

does not depend on the choice of reduced expression.

The relationship between symmetric and nonsymmetric Macdonald polynomials can then be stated as follows. By [5, (3.3.14)], for a partition $\lambda$ with at most $n$ parts, one has

[TABLE]

where $S_{n}^{\lambda}\subset S_{n}$ is the set of minimal coset representatives for $S_{n}/(S_{n})_{\lambda}$, where $(S_{n})_{\lambda}$ is the stabilizer of $\lambda$.

We note also, by [5, (3.3.40)], that whenever $\sigma_{i}(\mu)=\mu$, one has

[TABLE]

2.4 Partially symmetric Macdonald polynomials

Let $m$ be an integer such that $0\leq m\leq n$. Regard $S_{m}=S_{[m]}$ as the subgroup of $S_{n}$ fixing $m+1,\dotsc,n$. For $\lambda\in(\mathbb{Z}_{\geq 0})^{m}$ a partition with at most $m$ parts and $\gamma\in(\mathbb{Z}_{\geq 0})^{n-m}$ a composition, the partially symmetric Macdonald polynomial $P_{\lambda|\gamma}$ of [7, 13] is defined—following the the conventions of [7]—by

[TABLE]

where $\mu=(\lambda,\gamma)\in(\mathbb{Z}_{\geq 0})^{n}$. These form a $\mathbb{Q}(q,t)$-basis of $\mathbb{Q}(q,t)[x_{1},\dotsc,x_{n}]^{S_{m}}$, the algebra of polynomials which are symmetric in $x_{1},\dotsc,x_{m}$.

2.5 Permuted basement Macdonald polynomials

Our refinement of Theorem 1.1 will concern the summands of (2.5). For any $\pi\in S_{n}$ and $\mu\in(\mathbb{Z}_{\geq 0})^{n}$, define

[TABLE]

where $\ell_{\mu}(\pi)$ is the number of pairs $(i,j)\in[n]^{2}$ such that $i<j$, $\pi(i)>\pi(j)$, and $\mu_{i}\leq\mu_{j}$. In Remark 3.4 below, we will state the precise relationship between the $E_{\mu}^{\pi}$ and the permuted basement Macdonald polynomials of [1].

3 Combinatorics

In this section, we recall the combinatorial formula of Haglund, Haiman, and Loehr [9] for nonsymmetric Macdonald polynomials and its extension to permuted basement Macdonald polynomials due to Alexandersson [1]. Most of the following definitions come directly from [9] or from [1] (adapted to the conventions of [9]). We note that one difference arises, namely: when we define the inversion number $\operatorname{inv}(\widehat{\sigma})$ of an augmented filling $\widehat{\sigma}$, we will also take into account the permuted basements that we are allowing.

3.1 Column Diagrams

We may visualize a composition $\mu=(\mu_{1},\dotsc,\mu_{n})\in(\mathbb{Z}_{\geq 0})^{n}$ as a diagram of $n$ columns with $\mu_{i}$ boxes in column $i$. The column diagram of $\mu$ is the set

[TABLE]

in Cartesian coordinates. That is to say that $i$ indexes the columns and $j$ indexes the rows of the column diagram of $\mu$. Further, the augmented diagram of $\mu$ is given by

[TABLE]

where we simply adjoin $n$ boxes to row 0. For $\mu\in(\mathbb{Z}_{\geq 0})^{n}$ and $u=(i,j)\in\operatorname{dg}^{\prime}(\mu)$, we define

[TABLE]

and we set

[TABLE]

Example 3.1*.*

The diagram below depicts the diagram $\mu=(4,3,3,2,3,4)$ and the box $u=(3,2)$ with the cells belonging to $\text{leg}(u)$, $\text{arm}^{\text{left}}(u)$, and $\text{arm}^{\text{right}}(u)$ marked by $x$, $y$, and $z$, respectively. Here, $l(u)=1$ and $a(u)=2$.

u$$x$$y$$z

3.2 Fillings and Statistics

A filling of $\mu$ is a function

[TABLE]

where $[n]=\{1,2,\dotsc,n\}$. For a fixed permutation $\pi\in S_{n}$, the associated augmented filling is the associated filling $\widehat{\sigma}:\widehat{\operatorname{dg}}(\mu)\rightarrow[n]$ of the augmented diagram such that $\widehat{\sigma}$ agrees with $\sigma$ on $\operatorname{dg}^{\prime}(\mu)$, and $\widehat{\sigma}((j,0))=\pi(j)$ agrees with the given permutation for all $j\in[n]$. We will refer to the permutation $\pi$, or, equivalently row 0 of $\widehat{\sigma}$ as the basement of the augmented filling. Basements given by arbitrary permutations were considered in [1], but here they are adapted to the conventions of [9]. We refer the reader to Example 3.2 below for an illustration of our conventions.

Lattice squares $u,v\in(\mathbb{Z}_{\geq 0})^{2}$ such that $u\neq v$ are said to attack each other if either

they are in the same row (they have the form $(i,j),(i^{\prime},j))$, or 2. 2.

they are in consecutive rows, and the box in the lower row is to the right of the box in the upper row (they have the form $(i,j),(i^{\prime},j-1)$ where $i^{\prime}>i$).

A filling $\widehat{\sigma}:\widehat{\operatorname{dg}}(\mu)\rightarrow[n]$ is non-attacking if $\widehat{\sigma}(u)\neq\widehat{\sigma}(v)$ for each pair of attacking boxes $u,v\in\widehat{\operatorname{dg}}(\mu)$. We say that $\sigma:\operatorname{dg}(\mu)\rightarrow[n]$ is non-attacking if its associated augmented filling $\widehat{\sigma}$ is non-attacking. Following [1], for a composition $\mu$ and a permutation $\pi$, we denote the set of all non-attacking augmented fillings of $\mu$ with basement $\pi$ by $\mathrm{NAF}_{\mu}^{\pi}$. If $\pi$ is omitted, we assume that it is the identity permutation.

We write $d(u)=(i,j-1)$ to denote the box directly below the box $u=(i,j)$. A descent of an augmented filling $\widehat{\sigma}:\widehat{\operatorname{dg}}(\mu)\rightarrow[n]$ is a box $u\in\operatorname{dg}^{\prime}(\mu)$ such that $\widehat{\sigma}(u)>\widehat{\sigma}(d(u))$. Similarly, a box $u\in\operatorname{dg}^{\prime}(\mu)$ is an ascent if $\widehat{\sigma}(u)<\widehat{\sigma}(d(u)).$ For such a filling, we define

[TABLE]

The reading order on the set $\widehat{\operatorname{dg}}(\mu)$ is the total ordering of the boxes in $\widehat{\operatorname{dg}}(\mu)$ row by row, from top to bottom, and from right to left. In symbols, $(i,j)<(i^{\prime},j^{\prime})$ if $j<j^{\prime}$ or if $j=j^{\prime}$ and $i>i^{\prime}$. An inversion of a filling $\widehat{\sigma}:\widehat{\operatorname{dg}}(\mu)\rightarrow[n]$ is a pair of boxes $u,v\in\widehat{\operatorname{dg}}(\mu)$ such that $u,v$ attack each other, $u$ comes before $v$ in the reading order, and $\widehat{\sigma}(u)>\widehat{\sigma}(v).$ We define $\operatorname{Inv}(\widehat{\sigma})$ to be the set of all inversions and

[TABLE]

Example 3.2*.*

The diagram below depicts an augmented filling $\widehat{\sigma}\in\mathrm{NAF}_{(3,2,2,1,2,3)}^{132645}$, where the permutation $\pi=132645$ is written in one-line notation (so that $\pi(4)=6$ for instance).

132645132645132645****1

Recall that the bottom row is row 0. The descent set $\operatorname{Des}(\widehat{\sigma})$ consists of the two boxes bolded and colored in red, and the ascent set $\operatorname{Asc}(\widehat{\sigma})$ consists of the two boxes bolded and colored in blue. We find that $\operatorname{maj}(\widehat{\sigma})=2$. In row 2 we have 8 inversions, and in rows 1 and 0 we have 12 inversions each. Between consecutive rows, there are a total of 8 inversions, so $|\operatorname{Inv}(\widehat{\sigma})|=40$, which we encourage the reader to verify. Then we find that $\operatorname{inv}(\widehat{\sigma})=40-6-5=29$, and $\operatorname{coinv}(\widehat{\sigma})=32-29=3$.

3.3 Combinatorial formula

For $\pi=12\dotsm n$ equal to the identity, the main result of [9] is the combinatorial formula for the nonsymmetric Macdonald polynomials given below. In [1], this was extended to basements given by arbitrary permutations.

Theorem 3.3** ([1, 9]).**

The permuted basement Macdonald polynomials defined by (2.6) are given by the formula

[TABLE]

where $x^{\sigma}=\prod_{u\in\operatorname{dg}^{\prime}(\mu)}x_{\sigma(u)}.$

Remark 3.4*.*

Denoting by $A^{\pi}_{\mu}$ the permuted basement nonsymmetric Macdonald polynomials of [1] (where they are denoted $E^{\pi}_{\mu}$ but have different meaning), one has $E^{\pi}_{\mu}=A^{\pi\omega_{0}}_{\omega_{0}(\mu)}$, where $\omega_{0}\in S_{n}$ is the long element $\omega_{0}(i)=n+1-i$, the product $\pi\omega_{0}$ uses the group operation (composition) on $S_{n}$ as in this paper, and $\omega_{0}(\mu)=(\mu_{n},\dotsc,\mu_{1})$. This can be shown by induction on $\ell(\pi)$ using [1, Prop. 15].

It will be useful to reformulate $\operatorname{inv}(\widehat{\sigma})$ and $\operatorname{coinv}(\widehat{\sigma})$ as the number of inversion triples and coinversion triples, respectively. We say that a triple consists of three boxes $u,v,w\in\widehat{\operatorname{dg}}(\mu)$ such that

[TABLE]

Given a filling $\widehat{\sigma}$ and boxes $x,y$ in $\widehat{\operatorname{dg}}(\mu)$, with $x<y$ in the reading order, set

[TABLE]

If $(u,v,w)$ is a triple, then $v$ attacks $u$ and $w$, so we have $\chi_{uv}(\widehat{\sigma})=1$ if and only if $(u,v)\in\operatorname{inv}(\widehat{\sigma})$. Similarly, we have $\chi_{vw}(\widehat{\sigma})=1$ if and only if $(v,w)\in\operatorname{inv}(\widehat{\sigma})$, and we have $\chi_{uw}(\widehat{\sigma})=1$ if and only if $u\in\operatorname{Des}(\widehat{\sigma})$. It then follows that

[TABLE]

With that said, we say that a triple $(u,v,w)$ is an inversion triple of $\widehat{\sigma}$ if $\chi_{uv}(\widehat{\sigma})+\chi_{vw}(\widehat{\sigma})-\chi_{uw}(\widehat{\sigma})=1$, and we say that $(u,v,w)$ is a coinversion triple otherwise. We have the following useful Lemma regarding triples.

Lemma 3.5** ([9, Lemma 3.6.1]).**

Every pair of attacking boxes in $\widehat{\operatorname{dg}}(\mu)$ occurs as either $\{u,v\}$ or $\{v,w\}$ in a unique triple $(u,v,w)$, with the exception that an attacking pair $\{(i,0),(i^{\prime},0)\}$ in row [math] such that $i<i^{\prime}$ and $\mu_{i}\leq\mu_{i^{\prime}}$, does not belong to any triple.

We will now state and prove a Proposition from [9] since the proof differs slightly on account of our definition of $\operatorname{inv}(\widehat{\sigma})$.

Proposition 3.6** ([9], Proposition 3.6.2).**

Let $\sigma$ be a filling of $\mu$. The number of inversion triples (coinversion triples) of $\widehat{\sigma}$ is equal to $\operatorname{inv}(\widehat{\sigma})$ ($\operatorname{coinv}(\widehat{\sigma}))$.

Proof.

We claim that the number of inversion triples is given by the sum over all triples

[TABLE]

Naively, one might assume that the contribution to the sum from the first two terms comes from $|\operatorname{Inv}(\widehat{\sigma})|$; however, it follows from Lemma 3.5 that we must subtract from $|\operatorname{Inv}(\widehat{\sigma})|$ the number of inversions in the basement. Hence, the contribution to the sum from the first two terms is given by $|\operatorname{Inv}(\widehat{\sigma})|-|\{i<j:\mu_{i}\leq\mu_{j},\widehat{\sigma}(i,0)<\widehat{\sigma}(j,0)\}|$. Further, the contribution from the last term is simply given by $-\sum_{u\in\operatorname{Des}(\widehat{\sigma})}$, which proves that $\operatorname{inv}(\widehat{\sigma})$ is equal to the number of inversion triples. Further, we have exactly one triple $(u,v,w)$ for each $u\in\operatorname{dg}^{\prime}(\mu)$ and $v\in\text{arm}(u)$. Thus, the number of triples is equal to $\sum_{u\in\operatorname{dg}^{\prime}(\mu)}\text{arm}(u)$, so the number of coinversion triples is equal to $(\sum_{u\in\operatorname{dg}^{\prime}(\mu)}\text{arm}(u))-\operatorname{inv}(\widehat{\sigma})=\operatorname{coinv}(\widehat{\sigma})$, and we are done. ∎

It will be helpful to characterize triples pictorially. Figure 1 depicts Type I and Type II triples, assuming that the column to which $u$ belongs is strictly taller than the column to which $v$ belongs in the Type I case and weakly taller in the type II case.

We have the following useful description for coinversion triples:

Lemma 3.7** ([9], Lemma 3.6.3).**

If $\sigma$ is a non-attacking filling, then a triple $(u,v,w)$ is a coinversion triple if and only if $\widehat{\sigma}(u)<\widehat{\sigma}(v)<\widehat{\sigma}(v)$ or $\widehat{\sigma}(v)<\widehat{\sigma}(w)<\widehat{\sigma}(u)$ or $\widehat{\sigma}(w)<\widehat{\sigma}(u)<\widehat{\sigma}(v)$.

In other words, a triple is a coinversion triple if and only if its entries increase clockwise in Type 1 or counterclockwise in Type 2. Knowing this will be helpful in our later proofs. In [9] the authors recast their combinatorial formula as a formula for what they call the “opposite” Macdonald polynomials $E_{\mu}(x;q^{-1},t^{-1})$. They define

[TABLE]

where it will be helpful to realize that $\operatorname{maj}^{\prime}(\widehat{\sigma})=\operatorname{maj}(\widehat{\sigma}^{\prime})$ and $\operatorname{coinv}^{\prime}(\widehat{\sigma})=\operatorname{coinv}(\widehat{\sigma}^{\prime})$, where $\widehat{\sigma}^{\prime}(u)=n+1-\widehat{\sigma}(u)$. In the following section, this realization will prove to be especially useful. We end this section with the following Corollary of Theorem 3.3.

Corollary 3.8** ([9], Corollary 3.6.4).**

[TABLE]

3.4 Permuted basement identity

As noted in [1], Corollary 3.8 provides an interesting relationship between non-attacking fillings of a diagram with the identity permutation in the basement, and non-attacking fillings of a diagram with the reversed identity permutation in the basement. The following work will generalize this result, and provide a similar relationship between non-attacking fillings with different but comparable basements.

For a permutation $\pi\in S_{n}$, we write $\pi$ in the usual one-line notation $\pi=\pi(1)\pi(2)\dotsm\pi(n)$. Its complement, denoted $\pi^{c}$, is given by $\pi^{c}=(n+1-\pi(1))(n+1-\pi(2))\dotsm(n+1-\pi(n))$. For example, if $\pi=1432$, then $\pi^{c}=4123$. For the next theorem we will state, we will consider a permutation $\pi$ that can be written as the concatenation of two permutations of disjoint sets of consecutive integers. For example, the permutation $\pi=132546$ can be thought of as the concatenation of $\pi_{1}=132$ and $\pi_{2}=546$. For our purposes, we will consider the complements of $\pi_{1}$ and $\pi_{2}$ with respects to the sets on which they are permutations. That is, if $A=[a,b]=\{a,a+1,\dotsc,b-1,b\}$ is a finite set of consecutive positive integers where $1\leq a\leq b$ and $|A|=n$, then for any $\pi\in S_{A}$, we say

[TABLE]

Then with this definition of the complement, we see that for $\pi_{1}=132$ and $\pi_{2}=546$, we have $\pi_{1}^{c}=312$ and $\pi_{2}^{c}=564$. We will now state and work towards proving the following, which is our main result and the promised refinement of the Concha-Lapointe identity (cf. §4).

Theorem 3.9**.**

Let $\pi_{1}$ be a permutation of the set $[m]$ and $\pi_{2}$ be a permutation of the set $\{m+1,m+2,\dotsc,n\}$ for some $m,n\in\operatorname{\mathbb{N}}$ with $n>m$. Let $\pi=\pi_{1}\pi_{2}$ be the concatenation of these permutations. Let $p=\sum_{m+1\leq i\leq n}\mu_{i}$ Then

[TABLE]

To avoid triviality, assume that both $\pi_{1},\pi_{2}$ are nonempty. Before proving Theorem 3.9, we will prove three Lemmas to help us understand the underlying combinatorics behind the stated identity. With each Lemma, we will provide a pictorial example demonstrating the proof of each Lemma.

Lemma 3.10**.**

Let $\mu\in\operatorname{\mathbb{N}}^{n}$ be arbitrary, and let $\pi_{1}\in S_{m}$ and $\pi_{2}\in S_{\{m+1,\dotsc,n\}}$. Then

[TABLE]

Proof.

There exists a natural function $f:\mathrm{NAF}_{\mu}^{\pi_{1}\pi_{2}}\rightarrow\mathrm{NAF}_{\mu}^{\pi_{1}^{c}\pi_{2}^{c}}$ such that for any $\widehat{\sigma}\in\mathrm{NAF}_{\mu}^{\pi_{1}\pi_{2}}$, $f(\widehat{\sigma})$ is the filling defined on each box of $\operatorname{dg}^{\prime}(\mu)$ by

[TABLE]

for each $u\in\operatorname{dg}^{\prime}(\mu)$. Indeed, $f$ is well-defined. Without loss of generality, take a box $(i,0)$ in the basement of $\widehat{\sigma}$, such that $\widehat{\sigma}((i,0))\leq m$. Then we have

[TABLE]

and for any two boxes $u,v\in\operatorname{dg}^{\prime}(\mu)$, we have $\widehat{\sigma}(u)=\widehat{\sigma}(v)$ if and only if $f(\widehat{\sigma}(u))=f(\widehat{\sigma}(v))$, and since $\widehat{\sigma}$ is non-attacking, $f(\widehat{\sigma})$ is also non-attacking. Hence, $f(\widehat{\sigma})\in\mathrm{NAF}_{\mu}^{\pi_{1}^{c}\pi_{2}^{c}}.$ This function is clearly one-to-one and onto, so $f$ is a bijection, which proves the statement. ∎

Example 3.11*.*

The diagram below depicts the augmented filling $\widehat{\sigma}$ of a non-attacking filling $\sigma$ of $\mu=(3,2,2,1,2,3)$ and the corresponding filling $f(\widehat{\sigma})$.

$\widehat{\sigma}=$3124653124653124653,1326451326451326451$f(\widehat{\sigma})=$

Notice that $\widehat{\sigma}\in\mathrm{NAF}_{\mu}^{312465}$ and $f(\widehat{\sigma})\in\mathrm{NAF}_{\mu}^{132645}.$

Lemma 3.12**.**

Let $\pi_{1}$ be a permutation of the set $[m]$ and $\pi_{2}$ be a permutation of the set $\{m+1,m+2,\dotsc,n\}$ for some $m,n\in\operatorname{\mathbb{N}}$ with $n>m$. For a given composition $\mu\in\operatorname{\mathbb{N}}^{n}$, let $\widehat{\sigma}\in\mathrm{NAF}_{\mu}^{\pi_{1}\pi_{2}}$ be arbitrary. Further, let $L=|(f(\widehat{\sigma}))^{-1}(\{m+1,\dotsc,n\})|$, and let $p=\sum_{m+1\leq i\leq n}\mu_{i}$. Then

[TABLE]

Proof.

Informally speaking, $L$ represents the number of boxes of $\operatorname{dg}^{\prime}(\mu)$ that have fillings which are one of the entries of $\pi_{2}$ under the filling $f(\widehat{\sigma})$. We use $L$ to denote this number since these fillings are “large” compared to the entries of $\pi_{1}$. We will show that both sides of $(\ref{maj,maj'})$ count the same objects. First, we introduce a relation $\sim$ on the boxes of $\widehat{\operatorname{dg}}(\mu)$ under the filling $\widehat{\sigma}$. For any $u,v\in\widehat{\operatorname{dg}}(\mu)$, we say that $u\sim v$ if both $\widehat{\sigma}(u)$, $\widehat{\sigma}(v)\in[m]$ or both $\widehat{\sigma}(u)$, $\widehat{\sigma}(v)\in\{m+1,\dotsc,n\}$. Informally speaking, $u\sim v$ if $\widehat{\sigma}(u)$ and $\widehat{\sigma}(v)$ are both entries of $\pi_{1}$, the “small” entries, or both entries of $\pi_{2}$, the “large” numbers. By definition,

[TABLE]

which we may rewrite as

[TABLE]

Notice that by definition of $f$, we have that $u\in\operatorname{Des}(f(\widehat{\sigma}))$ and $u\sim d(u)$ if and only if $u\in\operatorname{Asc}(\widehat{\sigma})$ and $u\sim d(u)$. Similarly, $u\in\operatorname{Des}(f(\widehat{\sigma}))$ and $u\nsim d(u)$ if and only if $u\in\operatorname{Des}(\widehat{\sigma})$ and $u\nsim d(u)$. Hence, we may write

[TABLE]

which will we finally rewrite as

[TABLE]

Notice that the first term of the right-hand side of (3.7) is given by $\operatorname{maj}^{\prime}(\widehat{\sigma})$ by definition. Additionally, notice that $-p+L$ yields the number of boxes in the first $m$ columns with large numbers for fillings and the negative number of boxes in the last $n-m$ columns with small numbers for fillings. We will show that the last two terms of $(\ref{maj of fhsig})$ also count these boxes.

Start by taking any one of the first $m$ columns, and consider the empty space above the highest box in this column. Beginning at this box, we will go down the column and count boxes until we reach row 1. Because we begin at an imaginary box, it will not be included in our count. Once we see a box $u$ such that $d(u)$ has a filling with a large number, we will subtract the total number of boxes that we have seen up until this point. This yields the negative number of boxes that we have seen with small numbers for fillings. Now we continue until we see a box $u$ such that $d(u)$ has a small number for a filling, and at this point, we will add the total number of boxes that we have seen since the beginning. Note that such a box exists since the any box in the basement in the first $m$ columns has a small number for a filling. After adding, we will cancel out the number of boxes with small fillings that we have seen, and we will be left with the number of boxes with large fillings. Continuing on in this manner, each time we subtract, we will be left with the negative number of boxes that have small numbers for fillings, and each time we add, we will be left with the number of boxes with large numbers for fillings. As previously discussed, because the basement in the first $m$ columns strictly has small numbers for fillings, we will always add last throughout this process, leaving us with the total number of boxes in the column with large numbers for filling. Do this process for any of the first $m$ columns, and we now have the number of boxes in the first $m$ columns with large numbers for fillings. See the example following this proof for a demonstration of this process.

Turning our attention to the last $n-m$ columns, we can repeat the same process. However, because the basement in these columns strictly has large numbers for fillings, the last thing we will do throughout this process is subtract, which we know leaves us with the negative number of boxes in the column with small numbers for a filling. This process of moving down columns and adding and subtracting by the total number of boxes we have seen each time $u\nsim d(u)$ is given by

[TABLE]

which is what we wanted to show. ∎

Example 3.13*.*

For $\mu=(5,2,2,1,2,5)$, consider the the augmented filling of $\sigma\in\mathrm{NAF}_{\mu}^{312564}$ depicted below.

31256431256431264532165-0$$+1$$-2$$+3$$-0$$+1$$-3

On the left, we start from the imaginary box in the first column, and every time we have a box $u$ such that $d(u)$ is a large number, we subtract the total number of boxes seen. Since the first box we see is imaginary, we subtract 0. Then, once we reach a box $u$ such that $d(u)$ is a small number, we add the total number of boxes seen. In our example, we add 1 since the box with the filling 6 is the only real box we have seen. We continue on in this manner, and the final sum $0+1-2+3=2$ is the number of boxes in the first column that have a large number for a filling. Similarly, on the right, we see that $0+1-3=2$ is the negative number of boxes in the last column that have a small number for a filling.

Lemma 3.14**.**

Let $\pi_{1}$ be a permutation of the set $[m]$ and $\pi_{2}$ be a permutation of the set $\{m+1,m+2,\dotsc,n\}$ for some $m,n\in\operatorname{\mathbb{N}}$ $n>m$. Let $\widehat{\sigma}\in\mathrm{NAF}_{\mu}^{\pi_{1}\pi_{2}}$ be arbitrary. Then

[TABLE]

Proof.

Recall that $\operatorname{coinv}(f(\widehat{\sigma}))$ counts the number of coinversion triples of $f(\widehat{\sigma})$. Further, by Lemma 3.7, coinversion triples contain distinct entries. Without loss of generality, we will only consider coniversion triples of Type I as a similar argument holds for coinversion triples of Type II. In Type I, the entries of a coninversion triple must increase clockwise. Thus, without loss of generality, we may assume that $f(\widehat{\sigma}(u))<f(\widehat{\sigma}(v))<f(\widehat{\sigma}(w))$ for a coinversion triple $(u,v,w)$ of $f(\widehat{\sigma})$. If $f(\widehat{\sigma}(u))\sim f(\widehat{\sigma}(v))\sim f(\widehat{\sigma}(w))$, then we have $\widehat{\sigma}(u)>\widehat{\sigma}(v)>\widehat{\sigma}(w)$, so $(u,v,w)$ is an inversion triple with distinct entries in $\widehat{\sigma}$. If $f(\widehat{\sigma}(u))\nsim f(\widehat{\sigma}(v))\sim f(\widehat{\sigma}(w))$, then we must have $\widehat{\sigma}(v)>\widehat{\sigma}(w)>\widehat{\sigma}(u)$, so $(u,v,w)$ is an inversion triple with distinct entries in $\widehat{\sigma}$. Lastly, if $f(\widehat{\sigma}(u))\sim f(\widehat{\sigma}(v))\nsim f(\widehat{\sigma}(w))$, the we must have $\widehat{\sigma}(w)>\widehat{\sigma}(u)>\widehat{\sigma}(v)$, so $(u,v,w)$ is an inversion triple with distinct entries in $\widehat{\sigma}$. Similarly, an inversion triple of $\widehat{\sigma}$ with distinct entries always corresponds to a coinversion triple of $f(\widehat{\sigma})$. Hence, $\operatorname{coinv}(f(\widehat{\sigma}))=\operatorname{coinv}^{\prime}(\widehat{\sigma})$. ∎

Example 3.15*.*

Consider the two diagrams below, where the boxes in $f(\widehat{\sigma})$ that are bolded and blue form a coinversion triple of Type I, and the corresponding boxes in $\widehat{\sigma}$ that are bolded and blue form an inversion triple of Type I with distinct entries.

$\widehat{\sigma}=$31246531246531246\mathbf{3}$$\mathbf{4}53$\mathbf{5}$,13264513264513264\mathbf{1}$$\mathbf{6}$$\mathbf{5}$$1$$f(\widehat{\sigma})=

We are now ready to prove Theorem 3.9.

Proof (of Theorem 3.9).

Let $\widehat{\sigma}\in\mathrm{NAF}_{\mu}^{\pi_{1}\pi_{2}}$. It follows simply from the definition of $f$ that each occurrence of a box with filling $1\leq i\leq m$ is in one-to-one correspondence with an occurrence of a box with filling $m+1-i$. Similarly, each occurrence of a box with filling $m+1\leq j\leq n$ is in one-to-one correspondence with an occurrence of a box with filling $n+m+1-i$. This is why we make the substitution of variables on the right side of equation (3.4). By Lemma 3.14, it follows that $\operatorname{coinv}(f(\widehat{\sigma}))=\operatorname{coinv}^{\prime}(\widehat{\sigma})$, which is why we make the variable substitution $t=t^{-1}$ on the right side of equation (3.4). Similarly, by Lemma 3.12, $\operatorname{maj}(f(\widehat{\sigma}))=\operatorname{maj}^{\prime}(\widehat{\sigma})-p+L$. Therefore, we make the variable substitution $q=q^{-1}$ to account for $\operatorname{maj}^{\prime}(\widehat{\sigma})$ and multiply by $q^{-p}$ to account for the $-p$ term. Additionally, to each variable $x_{j}$ for $m+1\leq j\leq n$, we attach the variable $q$ to account for the $L$ term, which follows by definition of $L$. Summing over all $\widehat{\sigma}\in\mathrm{NAF}_{\mu}^{\pi_{1}\pi_{2}}$ yields the desired result. ∎

Throughout this section, we assumed that both $\pi_{1},\pi_{2}$ were nonempty. However, if we assume that $\pi_{2}$ is empty and let $p=0$ where $p$ is as defined in Theorem 3.9, we recover Corollary 3.8.

4 Recovering the partially symmetric identity

Now we will use Theorem 3.9 to prove the Concha-Lapointe identity stated in Theorem 1.1.

Let $0\leq m\leq n$. In the setting of Theorem 3.9, we assume:

• •

$\mu=\lambda|\gamma$ where $\lambda\in(\mathbb{Z}_{\geq 0})^{m}$ with $\lambda_{1}\geq\dotsm\geq\lambda_{m}$ and $\gamma\in(\mathbb{Z}_{\geq 0})^{n-m}$,

• •

$\pi_{1}$ belongs to $S_{m}=S_{[m]}\subset S_{n}$,

• •

$\pi_{2}$ is the identity element of $S_{[m+1,n]}$.

In this case, $\ell_{\mu}(\pi_{1})$ counts the number of pairs $(i,j)\in[m]^{2}$ such that $i<j,\pi_{1}(i)>\pi_{1}(j),$ and $\lambda_{i}=\lambda_{j}$.

Let $(S_{m})_{\lambda}$ denote the stabilizer of $\lambda$, which is a standard parabolic subgroup of $S_{m}$. Let $S_{m}^{\lambda}$ be the set of minimal coset representatives for $S_{m}/(S_{m})_{\lambda}$. For arbitrary $\pi_{1}\in S_{m}$, let $\underline{\pi_{1}}\in(S_{m})_{\lambda}$ be the minimal representative of the coset $\pi_{1}(S_{m})_{\lambda}$.

Lemma 4.1**.**

With the notation and assumptions above, one has the following:

For $\pi_{1}\in S_{m}^{\lambda}$, $\ell_{\mu}(\pi_{1})=0$ and hence $E_{\mu}^{\pi_{1}}=T_{\pi_{1}}E_{\mu}$. 2. 2.

For $\pi\in S_{m}$, $E_{\mu}^{\pi_{1}}=E_{\mu}^{\underline{\pi_{1}}}$.

Proof.

Since $\pi_{1}\in S_{m}$ and $\lambda$ is weakly decreasing, $\ell_{\mu}(\pi_{1})$ is equal to the number of pairs $(i,j)\in[m]^{2}$ such that $i<j$, $\pi(i)>\pi(j)$, and $\lambda_{i}=\lambda_{j}$. But membership in $S_{m}^{\lambda}$ is characterized by $\pi_{1}(i)<\pi_{1}(j)$ for all $(i,j)\in[m]^{2}$ such that $i<j$ and $\lambda_{i}=\lambda_{j}$. This establishes the first statement.

For the second statement, let $\pi_{1}\in S_{m}$ be arbitrary and write $\pi_{1}=\underline{\pi_{1}}\sigma$ where $\sigma\in(S_{m})_{\lambda}$. Then $\ell(\pi_{1})=\ell(\underline{\pi_{1}})+\ell(\sigma)$ and hence $T_{\pi_{1}}=T_{\underline{\pi_{1}}}T_{\sigma}$. Since $\sigma$ can be expressed as a product of simple transpositions fixing $\lambda$, we have $T_{\sigma}E_{\mu}=t^{\ell(\sigma)}E_{\mu}$ by (2.4). Hence

[TABLE]

The factorization $\pi_{1}=\underline{\pi_{1}}\sigma$ gives that for all $(i,j)\in[m]^{2}$ such that $i<j$ and $\lambda_{i}=\lambda_{j}$, $\pi_{1}(i)>\pi_{1}(j)$ if and only if $\sigma(i)>\sigma(j)$. Thus $\ell_{\mu}(\pi_{1})=\ell(\sigma)$ and we are done. ∎

Proof of Theorem 1.1.

Recall that $k=n-m$ and $\omega_{0}^{(k)}$ be the longest element of $S_{[m+1,n]}\cong S_{k}$. By the definition of $P_{\lambda|\gamma}$ and Lemma 4.1, we have

[TABLE]

and by the definition of $E^{\pi}_{\mu}$ it follows that

[TABLE]

Recall that $p=\sum_{i=m+1}^{n}\mu_{i}=\sum_{i=1}^{k}\gamma_{i}$. We now invoke Theorem 3.9 to obtain

[TABLE]

where we also use Lemma 4.1 for the last equality.

The quantity $\ell_{\mu}(\omega_{0}^{(k)})$ is the number of pairs $(i,j)\in[m+1,n]^{2}$ such that $i<j$ and $\mu_{i}\leq\mu_{j}$, which is nothing but $\ell(\omega_{0}^{(k)})-\text{inv}(\gamma)$. Therefore

[TABLE]

where we use the fact that $\pi_{1}\mapsto\underline{\pi_{1}^{c}}$ is a bijection from $S_{m}^{\lambda}$ to itself and, finally, the symmetry of $P_{\lambda|\gamma}$ in $x_{1},\dotsc,x_{m}$. ∎

5 Kazhdan-Lusztig involution

Let $\mathcal{M}=\mathbb{Q}(q,t^{1/2})[x_{1},\dotsc,x_{n}]$, where $t^{1/2}$ is a formal variable such that $(t^{1/2})^{2}=t$. As explained (in greater generality) in [11], the space $\mathcal{M}$ has the structure of maximal parabolic module for the (positive) affine Hecke algebra $\mathcal{H}_{n}$. This action is generated by the Demazure-Lusztig operators $T_{i}$ for $i=1\dotsc,n-1$ and the operators of multiplication by $x_{i}$ for $i=1,\dotsc,n$. Moreover, the Kazhdan-Lusztig involution $*$ on $\mathcal{H}_{n}$ induces the following $\mathbb{Q}$-linear involution on $\mathcal{M}$:

[TABLE]

where $\omega_{0}=\omega_{0}^{[n]}$ is the long element of $S_{n}$ and $f(q^{-1},t^{-1})$ denotes the image of $f$ under the $\mathbb{Q}$-algebra automorphism of $\mathcal{M}$ sending $q\mapsto q^{-1},t^{1/2}\mapsto t^{-1/2}$, and $x_{i}\mapsto x_{i}$ for $i=1,\dotsc,n$.

5.1 Nonsymmetric case

Theorem 5.1** ([11, Theorem 4.8]).**

For any $\mu=(\mu_{1},\dotsc,\mu_{n})\in(\operatorname{\mathbb{Z}}_{\geq 0})^{n}$, the normalized nonsymmetric Macdonald polynomial $\tilde{E}_{\mu}=t^{\frac{\mathrm{inv}(\mu)}{2}}E_{\mu}$ satisfies $\tilde{E}_{\mu}^{*}=\tilde{E}_{\mu}$.

We observe that $\tilde{E}_{\mu}^{*}=\tilde{E}_{\mu}$ is equivalent to the nonsymmetric case ($m=0$, $k=n$) of the Concha-Lapointe identity (1.1), namely:

[TABLE]

Here we use the fact that $E_{\mu}$ is homogeneous of degree $\mu_{1}+\dotsm+\mu_{n}$ to eliminate the $q$-shifts.

5.2 Partially symmetric case

For $0\leq m\leq n$, define

[TABLE]

It is well-known and straightforward to verify that $e_{m}^{*}=e_{m}$ and also that

[TABLE]

Therefore, we may restrict the Kazhdan-Lusztig involution to $e_{m}\mathcal{M}$, where it is given by:

[TABLE]

Now we are ready to prove Theorem 1.2 which gives the partially symmetric extension of Theorem 5.1.

Proof of Theorem 1.2.

We immediately deduce from Theorem 5.1 that $e_{m}\tilde{E}_{\mu}$ will be fixed under $*$ for any $\mu\in(\mathbb{Z}_{\geq 0})^{n}$. For $\mu=\lambda|\gamma$ with $\lambda\in(\mathbb{Z}_{\geq 0})^{n}$ weakly decreasing and $\gamma\in(\mathbb{Z}_{\geq 0})^{k}$, we have

[TABLE]

Since the scalar in front of $\tilde{P}_{\lambda|\gamma}$ is invariant under $t\mapsto t^{-1}$, we deduce that $\tilde{P}_{\lambda|\gamma}^{*}=\tilde{P}_{\lambda|\gamma}$. Equivalently,

[TABLE]

Since $\lambda$ is weakly decreasing, we have

[TABLE]

and this proves the first assertion of Theorem 1.2.

To connect with the Concha-Lapointe identity (1.1) we need to work out the left-hand side of (5.6). Using equation (5.5) for the Kazhdan-Lusztig involution, we have

[TABLE]

In order to pass from $T_{\omega_{0}}^{-1}$ to $T_{\omega_{0}^{(k)}}^{-1}$, we make use of the following eigenoperator for nonsymmetric Macdonald polynomials (cf. [5, Prop. 3.3.1]):

[TABLE]

where $\varpi_{m}$ stands for the $m$-th $GL_{n}$-fundamental weight and $\pi^{-1}$ is the operator

[TABLE]

The eigenvalues of $Y_{-\varpi_{m}}$ on $E_{\mu}$ for $\mu=\lambda|\gamma$ are given as follows:

[TABLE]

where

[TABLE]

Using (5.9), we can write (5.8) as

[TABLE]

It is straightforward to verify that

[TABLE]

where $i^{*}=n-i$, and hence

[TABLE]

for any $f=f(q,t)=f(x_{1},\dotsc,x_{n};q,t)\in\mathcal{M}$. Returning to (5.11) and using

[TABLE]

we can now write $P_{\lambda|\gamma}^{*}$ as

[TABLE]

Since $E_{\mu}$ is homogeneous of degree $\sum_{i=1}^{m}\lambda_{i}+\sum_{i=1}^{k}\gamma_{i}$ (and recall that $p=\sum_{i=1}^{k}\gamma_{i}$), we can write this as

[TABLE]

Applying (5.12) for $i=1,\dotsc,m-1$ and $\omega_{0}^{[m]}$ in place of $\omega_{0}$ and then that $e_{m}^{*}=e_{m}$, we arrive at

[TABLE]

Finally, we see that the equivalence of (1.1) and (1.2) boils down to the equation

[TABLE]

which we establish as follows. First, we use (5.7) to write the left-hand side of (5.13) as

[TABLE]

We then obtain the desired quantity $\ell(\omega_{0}^{[m+1,n]})-\text{inv}(\gamma)$ by comparing with (5.10) and canceling from $\ell(\omega_{0})$ all pairs $(i,j)\in[n]^{2}$ with $i<j$ except those with $i,j\in[m+1,n]$. ∎