Macdonald characters from a new formula for Macdonald polynomials

Houcine Ben Dali Université de Lorraine, CNRS, IECL, F-54000 Nancy Université de Paris, CNRS, IRIF, F-75006 Paris, France. [email protected] and Michele D’Adderio Università di Pisa
Dipartimento di Matematica
Largo Bruno Pontecorvo 5, 56127 Pisa
Italy [email protected]

Abstract.

We introduce a new operator $\bm{\Gamma}$ on symmetric functions, which enables us to obtain a creation formula for Macdonald polynomials. This formula provides a connection between the theory of Macdonald operators initiated by Bergeron, Garsia, Haiman and Tesler, and shifted Macdonald polynomials introduced by Knop, Lassalle, Okounkov and Sahi.

We use this formula to introduce a two-parameter generalization of Jack characters, which we call Macdonald characters. Finally, we provide a change of variables in order to formulate several positivity conjectures related to these generalized characters. Our conjectures extend some important open problems on Jack polynomials, including some famous conjectures of Goulden and Jackson.

1. Introduction

1.1. Jack and Macdonald polynomials

Jack polynomials are symmetric functions depending on one parameter $\alpha$ which have been introduced by Jack [Jac71]. The combinatorial analysis of Jack polynomials has been initiated by Stanley [Sta89] and a first combinatorial interpretation has been given by Knop and Sahi in terms of tableaux [KS97]. A second family of combinatorial objects related to Jack polynomials is given by maps, which are roughly graphs embedded in surfaces. This connection has first been observed in the conjectures of Goulden and Jackson [GJ96a] and important progress has recently been made in this direction [CD22, BDD23] with a first “topological expansion” of Jack polynomials in terms of maps.

Macdonald polynomials are symmetric polynomials introduced by Macdonald in 1989, which depend on two parameters $q$ and $t$ . Jack polynomials can be obtained from Macdonald polynomials by taking an appropriate limit. Several combinatorial results on Jack polynomials have been generalized to the Macdonald case, in particular, an interpretation in terms of tableaux was established in [HHL05]. However, no connection between Macdonald polynomials and maps is known, even conjecturally.

While Jack polynomials were one of the original inspiration for Macdonald polynomials, the two objects have diverged a bit in recent years, with Jack polynomials being most interesting to those who study probability and maps, while Macdonald polynomials have been studied more in the context of coinvariants and related algebraic geometry (like affine Spinger fibers, Hilbert schemes of points, knot invariants, etc.).

Our hope is to “reunite” Macdonald polynomials and Jack polynomials by showing how Macdonald polynomials are directly connected back to the work of Stanley, Goulden, Jackson, Lassalle, and others on Jack polynomials and their positivity properties.

As a first step towards this “reunification”, we introduce in the present article some new tools that make the parallel between the Jack and Macdonald stories more compelling.

First, we prove a creation formula (Eqs. 4 and 3) for Macdonald polynomials, inspired from the one used in [BDD23] to connect Jack polynomials to maps. Second, we use this formula to introduce a Macdonald analog of Jack characters (Section 1.4). Finally, we formulate a Macdonald version of some Jack conjectures, including Goulden and Jackson’s Matchings-Jack and $b$ -conjectures.

1.2. Symmetric functions and plethysm

Consider the graded algebra $\Lambda=\oplus_{r\geq 0}\Lambda^{(r)}$ of symmetric functions in the alphabet $(x_{1},x_{2},\dots)$ with coefficients in $\mathbb{Q}(q,t)$ . Let $p_{\lambda}$ and $h_{\lambda}$ denote the power-sum and the complete symmetric functions in $(x_{i})_{i\geq 1}$ , respectively. We use here a variable $u$ to keep track of the degree of the functions, and an extra variable $v$ ; all the functions considered are in $\Lambda[v]\llbracket u\rrbracket$ . Consider the Hall scalar product defined by $\langle p_{\mu},p_{\nu}\rangle=\delta_{\mu,\nu}z_{\mu}$ , where $z_{\mu}$ is a numerical factor, see Section 2.1. Let $f^{\perp}$ denote the adjoint of multiplication by $f\in\Lambda$ with respect to $\langle-,-\rangle$ .

We will use the plethystic notation: if $E(q,t,u,v,x_{1},x_{2},\dots)\in\Lambda[v]\llbracket u\rrbracket$ and $f\in\Lambda$ then $f[E]$ is the image of $f$ under the algebra morphism defined by

	$\displaystyle\Lambda[v]\llbracket u\rrbracket\longrightarrow$	$\displaystyle\Lambda[v]\llbracket u\rrbracket$
	$\displaystyle p_{k}\longmapsto$	$\displaystyle E(t^{k},q^{k},u^{k},v^{k},x_{1}^{k},\dots)\quad\text{ for every }k\geq 1.$

Set $X:=x_{1}+x_{2}+\dots$ . Notice that $f[X]=f(x_{1},x_{2},\cdots)$ for any $f$ . Moreover,

p_{k}\left[X\frac{1-q}{1-t}\right]=\frac{1-q^{k}}{1-t^{k}}p_{k}(x_{1},x_{2},\dots)\text{ and }p_{\lambda}[-X]=(-1)^{\ell(\lambda)}p_{\lambda}(x_{1},x_{2},\dots).

We consider the scalar product

\langle f[X],g[X]\rangle_{q,t}:=\left\langle f[X],g\left[X\frac{1-q}{1-t}\right]\right\rangle.

In particular

(1)

\langle p_{\mu}[X],p_{\nu}[X]\rangle_{q,t}=\delta_{\mu,\nu}z_{\mu}(q,t):=\delta_{\mu,\nu}z_{\mu}p_{\mu}\left[\frac{1-q}{1-t}\right].

Finally, let $\mathcal{P}_{Z}$ be the operator such that

\mathcal{P}_{Z}\cdot f[X]=\operatorname{Exp}[ZX]f[X],

i.e. the multiplication by the plethystic exponential

\operatorname{Exp}[ZX]:=\sum_{n\geq 0}h_{n}[ZX]=\sum_{\mu\in\mathbb{Y}}\frac{p_{\mu}[ZX]}{z_{\mu}},

(here $\mathbb{Y}$ denotes the set of integer partitions) and let

\mathcal{T}_{Z}:=\sum_{\mu\in\mathbb{Y}}\frac{p_{\mu}[Z]p_{\mu}^{\perp}}{z_{\mu}}

be the translation operator, so that $\mathcal{T}_{Z}\cdot f[X]=f[X+Z]$ . Note that $\mathcal{P}_{Z+Z^{\prime}}=\mathcal{P}_{Z}\cdot\mathcal{P}_{Z^{\prime}}$ .

1.3. A new formula for Macdonald polynomials

In [BGHT99], the authors introduced a remarkable family of diagonal operators on a modified version of Macdonald polynomials. These operators, known as nabla and delta operators (see Section 3.1), have a rich combinatorial structure and are closely related to the space of diagonal harmonics [Hai02, CM18, HRW18, DM22]. We consider an analog of these operators for the integral form of Macdonald polynomials¹¹1We use boldface symbols to distinguish these operators from their relatives from Section 3.1. (denoted $J^{(q,t)}_{\lambda}$ ); let $\bm{\nabla}$ and $\bm{\Delta}_{v}$ be the operators on symmetric functions defined by

(2)

\bm{\nabla}\cdot J^{(q,t)}_{\lambda}\!=\!(-1)^{|\lambda|}\left(\prod_{\Box\in\lambda}q^{a^{\prime}(\Box)}t^{-\ell^{\prime}(\Box)}\right)J^{(q,t)}_{\lambda},

\bm{\Delta}_{v}\cdot J^{(q,t)}_{\lambda}\!=\!\prod_{\Box\in\lambda}\left(1-v\cdot q^{a^{\prime}(\Box)}t^{-\ell^{\prime}(\Box)}\right)J^{(q,t)}_{\lambda},

where the products run over the cells of the Young diagram of $\lambda$ , and where $a^{\prime}$ and $\ell^{\prime}$ are defined in Section 2.1.

We also introduce the following operator²²2This operator is a close relative of the Theta operator in [DR23], first introduced in [DIVW21]. on $\Lambda[v]\llbracket u\rrbracket$

\bm{\Gamma}(u,v):=\bm{\Delta}_{1/v}\mathcal{P}_{\frac{uv(1-t)}{1-q}}\bm{\Delta}_{1/v}^{-1}\ \ .

The polynomiality of $\bm{\Gamma}(u,v)$ in the variable $v$ is a consequence of the Pieri rule.

We can now state our new formula for Macdonald polynomials.

Theorem 1.1.

For any partition $\lambda=[\lambda_{1},\lambda_{2},\dots,\lambda_{k}]$ , we have

(3)

\mathbf{\Gamma}^{(+)}_{\lambda_{1}}\mathbf{\Gamma}^{(+)}_{\lambda_{2}}\cdots\mathbf{\Gamma}^{(+)}_{\lambda_{k}}\cdot 1={J}^{(q,t)}_{\lambda}\quad\text{ where }\quad\mathbf{\Gamma}^{(+)}_{m}:=[u^{m}]\bm{\nabla}^{-1}\bm{\Gamma}(u,q^{m})\bm{\nabla}.

It turns out that 1.1 is an easy consequence of the following creation formula.

Theorem 1.2.

For any partition $\lambda=[\lambda_{1},\lambda_{2},\dots,\lambda_{k}]$ , we have

(4)

\displaystyle\bm{\Gamma}(u,q^{\lambda_{1}})\bm{\Gamma}(t^{-1}u,q^{\lambda_{2}})\cdots\bm{\Gamma}(t^{-(k-1)}u,q^{\lambda_{k}})\cdot 1

\displaystyle=t^{-n(\lambda)}\bm{\nabla}\mathcal{T}_{\frac{1}{u(1-t)}}{J}^{(q,t)}_{\lambda}[uX],

where $n(-)$ is a statistic on Young diagrams, see Section 2. In Section 3.4, we prove analogous results for modified Macdonald polynomials (cf. 3.2) from which we deduce Theorems 1.2 and 1.1.

In addition to giving a direct construction of Macdonald polynomials, 1.2 provides a dual approach to study the structure of these polynomials. Indeed, thank to Eq. 4 we can think of ${J}^{(q,t)}_{\lambda}$ as a function in the partition $\lambda$ described by the alphabet $(q^{\lambda_{1}},q^{\lambda_{2}},\dots)$ . This dual approach plays a key role in this paper and is used in Section 4 to introduce a $q,t$ -deformation of the characters of the symmetric group.

1.4. Shifted symmetric functions and Macdonald characters

Kerov and Olshanski have introduced in [KO94] a new approach to study the asymptotic of the characters of the symmetric group, in which the characters are thought of functions in Young diagrams. These functions are known to have a structure of shifted symmetric functions.

This approach has been generalized to the Jack case by Lassalle, who introduced Jack characters in [Las08]. The latter are directly related to the coefficients of Jack polynomials in the power-sum basis and have been useful to understand asymptotic behavior of large Young diagrams sampled with respect to a Jack deformed Plancherel measure [CDM23, DF16]. Recently, a combinatorial interpretation of Jack characters in terms of maps on non orientable surfaces has been proved in [BDD23], answering a positivity conjecture of Lassalle.

We extend here this investigation by introducing Macdonald characters. We start by recalling the definition of $(q,t)$ -shifted symmetric polynomials, due to Okounkov [Oko98].

Definition 1.3.

We say that a polynomial in $k$ variables $f(v_{1},\dots,v_{k})$ is (q,t)-shifted symmetric (or simply shifted symmetric) if it is symmetric in the variables $v_{1},v_{2}t^{-1},\dots,v_{k}t^{1-k}.$

A shifted symmetric function $f(v_{1},v_{2},\dots)$ is a sequence $(f_{k})_{k\geq 1}$ of polynomials of bounded degrees, such that for each $k$ the function $f_{k}$ is a shifted symmetric polynomial in $k$ variables and which satisfy the following compatibility property

f_{k+1}(v_{1},\dots,v_{k},1)=f_{k}(v_{1},\dots,v_{k}).

We now use the operator $\bm{\Gamma}$ to introduce a two parameter deformation $\bm{\widetilde{\theta}}^{(q,t)}_{\mu}$ of the characters of the symmetric group.

Definition 1.4.

Fix a partition $\mu$ and an integer $k\geq 1$ . The Macdonald character with $k$ variables associated to $\mu$ is the function $\bm{\widetilde{\theta}}^{(q,t)}_{\mu,k}(v_{1},v_{2},\dots,v_{k})$ defined by

(5)

\displaystyle\bm{\widetilde{\theta}}^{(q,t)}_{\mu,k}(v_{1},v_{2},\dots):=\left\langle p_{\mu},\bm{\Gamma}(1,v_{1})\bm{\Gamma}(t^{-1},v_{2})\cdots\bm{\Gamma}(t^{-{k-1}},v_{k})\cdot 1\right\rangle_{q,t}

It turns out that these characters have a structure of shifted symmetric functions.

Theorem 1.5.

Fix a partition $\mu$ . For any $k\geq 1$ , the character $\bm{\widetilde{\theta}}^{(q,t)}_{\mu,k}$ is a shifted symmetric polynomial. Moreover, the sequence $(\bm{\widetilde{\theta}}^{(q,t)}_{\mu,k})_{k\geq 1}$ defines a shifted symmetric function $\bm{\widetilde{\theta}}^{(q,t)}_{\mu}$ , which will be called the Macdonald character associated to the partition $\mu$ .

Taking an appropriate limit (cf. 5.4), one can recover Jack characters from Macdonald characters, and hence also the characters of the symmetric group.

It follows from 3.2 that Macdonald characters are directly related to the expansion of Macdonald polynomials in the power-sum basis. We use the creation formula of 1.2 to deduce properties of the Macdonald characters which generalize results known in the Jack case.

In particular, we prove that they form a basis of the space of shifted symmetric functions which lead us to introduce their structure coefficients $\bm{g}^{\pi}_{\mu,\nu}$ , see 4.4 and Eq. 44. We also make a connection between Macdonald characters and shifted Macdonald polynomials introduced in [Las98, Oko98], see Equations (24) and (28).

Furthermore, Macdonald characters and their structure coefficients seem to have interesting positivity properties which we investigate by introducing a new parametrization for Macdonald polynomials.

1.5. A new parametrization and Macdonald version of some Jack conjectures

In Section 5.1 we introduce a new parametrization $(\alpha,\gamma)$ for Macdonald polynomials which is related to Jack polynomials, see Eq. 35. We show that this paremetrization gives a natural way to give a Macdonald version of some famous conjectures about Jack polynomials. In particular, we formulate two positivity conjectures about Macdonald characters $\bm{\widetilde{\theta}}^{(q,t)}_{\mu}$ (see 4) and their structure coefficients (see 9). These conjectures suggest that the characters $\bm{\widetilde{\theta}}^{(q,t)}_{\mu}$ have a combinatorial structure which generalizes the one given by maps and that we hope to investigate in future works. We also provide a Macdonald generalization of Stanley’s conjecture about the structure coefficients of Jack polynomials, see 3.

In [GJ96a], Goulden and Jackson introduced two conjectures which suggest that two families of coefficients, $c^{\pi}_{\mu,\nu}(\alpha)$ and $h^{\pi}_{\mu,\nu}(\alpha)$ , obtained from the expansion of some Jack series satisfy integrality and positivity properties. These conjectures, known as the Matching-Jack conjecture and the $b$ -conjecture, have also a combinatorial interpretation related to counting weighted maps on non-orientable surfaces.

The Matching-Jack and the $b$ -conjectures are still open despite many partial results [DF16, DF17, CD22, BD22, BD23]. These works involve various techniques including representation theory, random matrices and differential equations.

In Section 5.4, we introduce two families of coefficients $\bm{c}^{\pi}_{\mu,\nu}(\alpha,\gamma)$ and $\bm{h}^{\pi}_{\mu,\nu}(\alpha,\gamma)$ generalizing the coefficients of the Matchings-Jack and the $b$ -conjecture. These coefficients are obtained from the expansion of some Macdonald series with the parametrization $(\alpha,\gamma)$ , see Eqs. 41 and 42.

It turns out that the coefficients $\bm{c}^{\pi}_{\mu,\nu}$ are a special case of the structure coefficients of Macdonald characters $\bm{\widetilde{\theta}}^{(q,t)}_{\mu}$ , see 5.9. We also establish in 5.10 a connection between these coefficients and the super nabla operator recently introduced in [BHIR23].

We hope that our Macdonald generalization of these conjectures could reveal new combinatorial structures of Macdonald polynomials, in particular in connection with the enumeration of maps. Moreover, generalizing the open problems about Jack polynomials (the $b$ -conjecture, Stanley’s conjecture…) to the Macdonald setting might provide a new point of view to approach these conjectures, and give the possibility to use the tools provided by the theory of Macdonald polynomials, which do not all have interesting analogues in the Jack story.

1.6. Outline of the paper

The paper is organized as follows. In Section 2, we give some preliminaries and notation related to partitions and Macdonald polynomials. We prove the main theorem in Section 3. We introduce Macdonald characters in Section 4. We introduce several conjectures related to these characters in Section 5.

1.7. Acknowledgments

The first author was partially supported by the LUE DrEAM project of Université de Lorraine. He is also grateful to Valentin Féray and Guillaume Chapuy for many enlightening discussions.

The second author was partially supported by PRIN 2022A7L229 ALTOP, INDAM research group GNSAGA, and ARC grant “From algebra to combinatorics, and back”.

The authors are happy to thank the anonymous referee of an extended abstract of the present work for useful comments.

2. Preliminaries

For the results of this section we refer to [DR23, Mac95].

2.1. Partitions

A partition $\lambda=[\lambda_{1},...,\lambda_{\ell}]$ is a weakly decreasing sequence of positive integers $\lambda_{1}\geq...\geq\lambda_{\ell}>0$ . We denote by $\mathbbm{Y}$ the set of all integer partitions. The integer $\ell$ is called the length of $\lambda$ and is denoted $\ell(\lambda)$ . The size of $\lambda$ is the integer $|\lambda|:=\lambda_{1}+\lambda_{2}+...+\lambda_{\ell}.$ If $n$ is the size of $\lambda$ , we say that $\lambda$ is a partition of $n$ and we write $\lambda\vdash n$ . The integers $\lambda_{1}$ ,…, $\lambda_{\ell}$ are called the parts of $\lambda$ . For $i\geq 1$ , we denote $m_{i}(\lambda)$ the number of parts of size $i$ in $\lambda$ . We then set

z_{\lambda}:=\prod_{i\geq 1}m_{i}(\lambda)!i^{m_{i}(\lambda)}.

We denote by $\leq$ the dominance partial ordering on partitions, defined by

\mu\leq\lambda\iff|\mu|=|\lambda|\hskip 8.5359pt\text{ and }\hskip 8.5359pt\mu_{1}+...+\mu_{i}\leq\lambda_{1}+...+\lambda_{i}\text{ for }i\geq 1,

where we set $\lambda_{j}=0$ for $j>\ell(\lambda)$ .

We identify a partition $\lambda$ with its Young diagram³³3One should picture $(i,j)\in\lambda$ as being a square box in the $(i,j)$ -entry of a matrix, as it is custom in the English notation of tableaux., defined by

\lambda:=\{(i,j)\mid 1\leq i\leq\ell(\lambda),1\leq j\leq\lambda_{i}\}.

The conjugate partition of $\lambda$ , denoted $\lambda^{\prime}$ , is the partition associated to the Young diagram obtained by reflecting the diagram of $\lambda$ with respect to the line $j=i$ :

\lambda^{\prime}:=\{(i,j)\mid 1\leq j\leq\ell(\lambda),1\leq i\leq\lambda_{j}\}.

Fix a cell $\Box:=(i,j)\in\lambda$ . Its arm and leg are respectively given by

a_{\lambda}(\Box):=|\{(i,c)\in\lambda,c>j\}|=\lambda_{i}-j,\text{ and }\ell_{\lambda}(\Box):=|\{(r,j)\in\lambda,r>i\}|=(\lambda^{\prime})_{j}-i.

Similarly, the co-arm and co-leg are defined by

a^{\prime}_{\lambda}(\Box):=|\{(i,c)\in\lambda,c<j\}|=j-1,\text{ and }\ell^{\prime}_{\lambda}(\Box):=|\{(r,j)\in\lambda,r<i\}|=i-1.

Finally, let the statistic $n$ on Young diagram

n(\lambda):=\sum_{1\leq i\leq\ell(\lambda)}\lambda_{i}(i-1)=\sum_{\Box\in\lambda}\ell_{\lambda}^{\prime}(\Box).

2.2. Integral form of Macdonald polynomials:

For the results in this section we refer to [Mac95, Chapter VI] and [GT96].

Macdonald has established the following characterization theorem for Macdonald polynomials.

Theorem 2.1.

The Macdonald polynomials ${J}^{(q,t)}_{\lambda}$ are the unique family of symmetric functions such that

•

For any $\lambda$ ,

${J}^{(q,t)}_{\lambda}=\sum_{\mu\leq\lambda}v^{\lambda}_{\mu}m_{\mu},$

for some coefficients $v^{\lambda}_{\mu}$ .

•

For any partitions $\lambda$ and $\rho$ ,

\left\langle{J}^{(q,t)}_{\lambda},{J}^{(q,t)}_{\rho}\right\rangle_{q,t}=\delta_{\lambda,\rho}{j}^{(q,t)}_{\lambda},

where

(6)

{j}^{(q,t)}_{\lambda}:=\prod_{\Box\in\lambda}\left(1-q^{a_{\lambda}(\Box)+1}t^{\ell_{\lambda}(\Box)}\right)\left(1-q^{a_{\lambda}(\Box)}t^{\ell_{\lambda}(\Box)+1}\right)

Moreover, for every $r\in\mathbb{N}$ the set $\{{J}^{(q,t)}_{\lambda}\mid\lambda\vdash r\}$ is a basis of $\Lambda^{(r)}$ .

If $Y:=y_{1}+y_{2}+\dots$ is a second alphabet of variables, then Cauchy identity for Macdonald polynomials reads (cf. Eq. 1)

(7)

\sum_{\lambda\vdash m}\frac{{J}^{(q,t)}_{\lambda}[X]{J}^{(q,t)}_{\lambda}[Y]}{{j}^{(q,t)}_{\lambda}}=\sum_{\pi\vdash m}\frac{p_{\pi}[X]p_{\pi}[Y]}{z_{\pi}(q,t)},\text{ for any $m\geq 0$.}

Moreover, we have the following plethystic substitution formula for Macdonald polynomials.

Theorem 2.2.

{J}^{(q,t)}_{\lambda}\left[\frac{1-v}{1-t}\right]=\prod_{\Box\in\lambda}\left(t^{\ell_{\lambda}^{\prime}(\Box)}-vq^{a^{\prime}_{\lambda}(\Box)}\right)

Let $\lambda$ be a partition. We write $\lambda\subseteq\xi$ if the diagram $\lambda$ is contained in the diagram of $\xi$ . Moreover, we say that $\xi/\lambda$ is a horizontal strip if $\lambda\subseteq\xi$ and in each column there is at most one cell in $\xi$ and not in $\lambda$ . In other terms, for every $i\geq 1$

\lambda^{\prime}_{i}\leq\xi^{\prime}_{i}\leq\lambda^{\prime}_{i}+1.

Theorem 2.3 (Pieri rule).

Let $\lambda$ be a partition and $k\geq 1$ . Then,

h_{k}\left[X\frac{1-t}{1-q}\right]\cdot{J}^{(q,t)}_{\lambda}=\sum_{\xi}\eta_{\lambda,\xi}{J}^{(q,t)}_{\xi},

for some coefficients $\eta_{\lambda,\xi}$ , where the sum is taken over partitions $\xi$ such that $\xi/\lambda$ is a horizontal strip of size $k$ . More generally, if $f$ is a symmetric function of degree $k$ then

f\cdot{J}^{(q,t)}_{\lambda}=\sum_{\lambda\subset_{k}\xi}d^{f}_{\lambda,\xi}{J}^{(q,t)}_{\xi},

for some coefficients $d^{f}_{\lambda,\xi}$ , where the sum is taken over the partitions $\xi$ obtained by adding $k$ cells to $\lambda$ .

3. A new creation formula for Macdonald polynomials

In this section we prove two creation formulas for modified Macdonald polynomials (Theorems 3.3 and 3.2) from which we deduce the creation formulas for the integral forms stated in the introduction.

3.1. Modified Macdonald polynomials

In [GH93], Garsia and Haiman introduced a modified version of Macdonald polynomials

\widetilde{H}^{(q,t)}_{\lambda}=t^{n(\lambda)}J_{\lambda}^{(q,1/t)}\left[\frac{X}{1-1/t}\right].

The operators $\nabla$ and $\Delta_{v}$ are respectively defined by

(8)

\nabla\widetilde{H}^{(q,t)}_{\lambda}:=(-1)^{|\lambda|}\prod_{\Box\in\lambda}q^{a_{\lambda}^{\prime}(\Box)}t^{\ell_{\lambda}^{\prime}(\Box)}\widetilde{H}^{(q,t)}_{\lambda},

(9)

\qquad\Delta_{v}\widetilde{H}^{(q,t)}_{\lambda}:=\prod_{\Box\in\lambda}\left(1-vq^{a_{\lambda}^{\prime}(\Box)}t^{\ell_{\lambda}^{\prime}(\Box)}\right)\widetilde{H}^{(q,t)}_{\lambda}.

These operators are related by the five-term relation of Garsia and Mellit [GM19]

(10)

\nabla\mathcal{P}_{\frac{u}{M}}\nabla^{-1}\mathcal{P}_{\frac{uv}{M}}=\Delta_{1/v}\mathcal{P}_{\frac{uv}{M}}\Delta_{1/v}^{-1}\ ,

where $M:=(1-q)(1-t)$ . Let

B_{\lambda}:=\sum_{\Box\in\lambda}q^{a^{\prime}_{\lambda}(\Box)}t^{\ell^{\prime}_{\lambda}(\Box)}=\sum_{1\leq i\leq\ell(\lambda)}t^{i-1}\frac{1-q^{\lambda_{i}}}{1-q},

and $D_{\lambda}:=MB_{\lambda}-1.$ We state another fundamental identity for Macdonald polynomials, due to Garsia, Haiman and Tesler [GHT01]: for any partition $\lambda$

(11)

\nabla\mathcal{P}_{-\frac{u}{M}}\mathcal{T}_{\frac{1}{u}}\widetilde{H}_{\lambda}[uX]=\operatorname{Exp}\left[-\frac{uXD_{\lambda}}{M}\right].

Remark 1.

Actually, a first connection between this identity and shifted Macdonald polynomials has been made in [GHT01, Theorem 3.2]. In the following, we prove that this identity can be "decomposed" using the operator $\Gamma$ . This reformulation is a key step to obtain the construction formula 1.1.

3.2. Creation formula for modified Macdonald polynomials

We start by proving a modified version of 1.2. Set

\Gamma(u,v):=\Delta_{1/v}\mathcal{P}_{\frac{uv}{1-q}}\Delta_{1/v}^{-1}.

Remark 2.

Consider the operator $\tilde{\Theta}(z;v):=\Delta_{v}\mathcal{P}_{-\frac{z}{M}}\Delta_{v}^{-1}$ introduced in [DIVW21]. Then this operator is related to $\Gamma$ by

\Gamma(u,v)=\tilde{\Theta}(uv;1/v)^{-1}\tilde{\Theta}(tuv;1/v).

Before proving the main theorem of this subsection, we start by establishing a second expression for the operator $\Gamma$ .

Lemma 3.1.

We have

\Gamma(u,v)=\nabla\mathcal{P}_{\frac{u}{M}}\nabla^{-1}\mathcal{P}_{\frac{uv}{1-q}}\nabla\mathcal{P}_{\frac{-tu}{M}}\nabla^{-1}.

Proof.

The operator $\Gamma$ can be rewritten as follows

\Gamma(u,v)=\left(\Delta_{1/v}\mathcal{P}_{\frac{uv}{M}}\Delta_{1/v}^{-1}\right)\left(\Delta_{1/v}\mathcal{P}_{\frac{-tuv}{M}}\Delta_{1/v}^{-1}\right)=\left(\Delta_{1/v}\mathcal{P}_{\frac{uv}{M}}\Delta_{1/v}^{-1}\right)\left(\Delta_{1/v}\mathcal{P}_{\frac{tuv}{M}}\Delta_{1/v}^{-1}\right)^{-1}.

Using the five-term relation (10) on each one of the two factors, we obtain

\Gamma(u,v)=\nabla\mathcal{P}_{\frac{u}{M}}\nabla^{-1}\mathcal{P}_{\frac{uv}{M}}\mathcal{P}_{\frac{-utv}{M}}\nabla\mathcal{P}_{\frac{-tu}{M}}\nabla^{-1}=\nabla\mathcal{P}_{\frac{u}{M}}\nabla^{-1}\mathcal{P}_{\frac{uv}{1-q}}\nabla\mathcal{P}_{\frac{-tu}{M}}\nabla^{-1}.\qed

We now prove the first creation formula for modified Macdonald polynomials.

Theorem 3.2.

For $\lambda=[\lambda_{1},\lambda_{2},\dots,\lambda_{k}]$ partition we have

(12)

\displaystyle\Gamma(u,q^{\lambda_{1}})\Gamma(tu,q^{\lambda_{2}})\cdots\Gamma(t^{\ell-1}u,q^{\lambda_{\ell}})\cdot 1

\displaystyle=\nabla\mathcal{T}_{\frac{1}{u}}\widetilde{H}_{\lambda}[uX]=\nabla\widetilde{H}_{\lambda}[uX+1].

Proof.

It follows, using 3.1, that

\displaystyle\Gamma(u,v_{1})\Gamma(tu,v_{2})\cdots\Gamma(t^{k-1}u,v_{k})\cdot 1

\displaystyle=\nabla\mathcal{P}_{\frac{u}{M}}\nabla^{-1}\mathcal{P}_{\frac{uv_{1}}{1-q}}\mathcal{P}_{\frac{utv_{2}}{1-q}}\cdots\mathcal{P}_{\frac{ut^{k-1}v_{k}}{1-q}}\nabla\mathcal{P}_{\frac{-ut^{k}}{M}}\nabla^{-1}\cdot 1\ \ .

Using $\nabla^{-1}\cdot 1=1$ and $\nabla\mathcal{P}_{-\frac{z}{M}}\cdot 1=\mathcal{P}_{\frac{z}{M}}\cdot 1$ (see e.g. [DR23, Eq. (1.47)] with $k=n$ ), we get

	$\displaystyle\Gamma(u,v_{1})\Gamma(tu,v_{2})\cdots\Gamma(t^{k-1}u,v_{k})\cdot 1$	$\displaystyle=\!\nabla\mathcal{P}_{\frac{u}{M}}\nabla^{-1}\mathcal{P}_{\frac{uv_{1}}{1-q}}\mathcal{P}_{\frac{utv_{2}}{1-q}}\cdots\mathcal{P}_{\frac{ut^{k-1}v_{k}}{1-q}}\mathcal{P}_{\frac{ut^{k}}{M}}\cdot 1$
(13)			$\displaystyle=\!\nabla\mathcal{P}_{\frac{u}{M}}\!\nabla^{-1}\operatorname{Exp}\!\!\left[\frac{ut^{k}X}{M}+\frac{uX}{1-q}\sum_{1\leq i\leq k}t^{i-1}v_{i}\right]$
		$\displaystyle=\!\nabla\mathcal{P}_{\frac{u}{M}}\!\nabla^{-1}\operatorname{Exp}\!\!\left[\frac{uX}{M}-\frac{uX}{M}(1-t)\sum_{1\leq i\leq k}t^{i-1}(1-v_{i})\right]\!.$

Fix now a partition $\lambda$ . Applying the previous equation, we get

	$\displaystyle\Gamma(u,q^{\lambda_{1}})\Gamma(tu,q^{\lambda_{2}})\cdots\Gamma(t^{\ell-1}u,q^{\lambda_{\ell}})\cdot 1$	$\displaystyle=\!\nabla\mathcal{P}_{\frac{u}{M}}\!\nabla^{-1}\operatorname{Exp}\!\!\left[\frac{uX}{M}-\frac{uX}{M}(1-t)\sum_{i\geq 1}t^{i-1}(1-q^{\lambda_{i}})\right]$
		$\displaystyle=\!\nabla\mathcal{P}_{\frac{u}{M}}\!\nabla^{-1}\operatorname{Exp}\!\!\left[-\frac{uXD_{\lambda}}{M}\right],$

Applying Eq. 11 concludes the proof of the theorem. ∎

3.3. Vanishing property and second creation formula

The purpose of this subsection is to prove a version of 1.1 for modified Macdonald polynomials.

Theorem 3.3.

For any partition $\lambda=[\lambda_{1},\lambda_{2},\dots,\lambda_{k}]$ , we have

(14)

{\Gamma}^{(+)}_{\lambda_{1}}{\Gamma}^{(+)}_{\lambda_{2}}\cdots{\Gamma}^{(+)}_{\lambda_{k}}\cdot 1=\widetilde{H}^{(q,t)}_{\lambda}

where

{\Gamma}^{(+)}_{m}:=[u^{m}]\nabla^{-1}\Gamma(u,q^{m})\nabla.

We start by stating the Pieri rule for modified Macdonald polynomials, which can be deduced from 2.3 (see [GT96, Proposition 2.7]).

Theorem 3.4 (Pieri rule for $\widetilde{H}^{(q,t)}_{\lambda}$ ).

Let $\lambda$ be a partition and $k\geq 1$ . Then,

h_{k}\left[\frac{X}{1-q}\right]\cdot\widetilde{H}^{(q,t)}_{\lambda}=\sum_{\xi}\widetilde{\eta}_{\lambda,\xi}\widetilde{H}^{(q,t)}_{\xi},

for some coefficients $\eta_{\lambda,\xi}$ , where the sum is taken over partitions $\xi$ such that $\xi/\lambda$ is a horizontal strip of size $k$ .

Combining 3.4 and the definition of $\Delta_{v}$ (see Eq. 9), we deduce the following formula;

(15)

\Gamma(u,v)\cdot\widetilde{H}^{(q,t)}_{\lambda}=\sum_{k\geq 0}u^{k}\sum_{\xi\vdash k+|\lambda|}\widetilde{\eta}_{\lambda,\xi}\widetilde{H}^{(q,t)}_{\xi}\prod_{\Box\in\xi/\lambda}\left(v-q^{a^{\prime}(\Box)}t^{\ell^{\prime}(\Box)}\right),

where the second sum is taken over partitions $\xi$ such that $\xi/\lambda$ is a horizontal strip of size $k$ . We now consider for each non-negative integer $s$ , the subspace of $\Lambda$ defined by

\Lambda_{\leq s}:=\text{Span}_{\mathbb{Q}(q,t)}\left\{\widetilde{H}^{(q,t)}_{\lambda},\text{ for partitions }\lambda\text{ such that }\lambda_{1}\leq s\right\}.

Remark 3.

Let $\Box_{0}$ the cell of coordinates $(1,s+1)$ . This cell is characterized by $a^{\prime}(\Box_{0})=s$ and $\ell^{\prime}(\Box_{0})=0$ . Notice that the condition $\lambda_{1}\leq s$ is equivalent to saying that $\Box_{0}$ is not a cell of the Young diagram of $\lambda$ .

We then have the following proposition.

Proposition 3.5.

Fix $s,k\geq 0$ two integers.

(1)

The space $\Lambda_{\leq s}$ is stabilized by the action of the operator $[u^{k}]\Gamma(u,q^{s})$ .
(2)

If $k>s$ , then $[u^{k}]\Gamma(u,q^{s})=0$ as operators on $\Lambda_{\leq s}$ .

Proof.

Fix a partition $\lambda$ such that $\lambda_{1}\leq s$ . From Eq. 15, we know that

(16)

[u^{k}]\Gamma(u,q^{s})\cdot\widetilde{H}^{(q,t)}_{\lambda}=\sum_{\xi\vdash k+|\lambda|}\widetilde{\eta}_{\lambda,\xi}\widetilde{H}^{(q,t)}_{\xi}\prod_{\Box\in\xi/\lambda}\left(q^{s}-q^{a^{\prime}(\Box)}t^{\ell^{\prime}(\Box)}\right),

where the sum is taken over horizontal strips $\xi/\lambda$ of size $k$ . Notice that, with the notation of 3, the quantity $q^{s}-q^{a^{\prime}(\Box)}t^{\ell^{\prime}(\Box)}$ is 0 if and only if $\Box=\Box_{0}$ . Moreover, if $\xi$ is a partition such that $\xi_{1}>s$ , then $\Box_{0}\in\xi/\lambda$ and therefore, $\xi$ does not contribute to the sum in Eq. 16. This proves $(1)$ . If $k>s$ then any horizontal strip $\xi/\lambda$ of size $k>s$ contains necessarily the cell $\Box_{0}$ . This gives (2). ∎

We are now ready to prove 3.3.

Proof of 3.3.

Let $m$ and $\ell$ denote respectively the size and the length of $\lambda$ . Extracting the coefficient of $u^{m}$ in Eq. 12, we get

(17)

\displaystyle\sum_{m_{1}+\dots+m_{\ell}=m}\left([u^{m_{1}}]\Gamma(u,q^{\lambda_{1}})\right)\cdots\left([u^{m_{\ell}}]\Gamma(t^{\ell-1}u,q^{\lambda_{\ell}}))\right)\cdot 1

\displaystyle=\nabla\widetilde{H}^{(q,t)}_{\lambda}[X].

Hence,

(18)

\displaystyle\sum_{m_{1}+\dots+m_{\ell}=m}\left([u^{m_{1}}]\nabla^{-1}\Gamma(u,q^{\lambda_{1}})\nabla\right)\cdots\left([u^{m_{\ell}}]\nabla^{-1}\Gamma(t^{\ell-1}u,q^{\lambda_{\ell}})\nabla\right)\cdot 1

\displaystyle=\widetilde{H}^{(q,t)}_{\lambda}[X].

We want to prove that the only tuple $(m_{1},\dots,m_{\ell})$ which contributes to this sum is $(\lambda_{1},\dots,\lambda_{\ell}).$ First, notice that for each tuple $(m_{1},\dots,m_{\ell})$ , we obtain by induction and using item (1) of 3.5 that for any $0\leq i\leq\ell$ we have

(19)

\left([u^{m_{\ell-i}}]\nabla^{-1}\Gamma(t^{\ell-i-1}u,q^{\lambda_{\ell-i}})\nabla\right)\cdots\left([u^{m_{\ell}}]\nabla^{-1}\Gamma(t^{\ell-1}u,q^{\lambda_{\ell}})\nabla\right)\cdot 1\in\Lambda_{\leq\lambda_{\ell-i}}.

Moreover, if for some $1\leq i\leq\ell$ , we have $m_{i}>\lambda_{i}$ , then from 3.5 item (2) and Eq. 19 we have

(20)

\left([u^{m_{i}}]\nabla^{-1}\Gamma(t^{i-1}u,q^{\lambda_{i}})\nabla\right)\cdots\left([u^{m_{\ell}}]\nabla^{-1}\Gamma(t^{\ell-1}u,q^{\lambda_{\ell}})\nabla\right)\cdot 1=0.

We deduce that any tuple $(m_{1},\dots,m_{\ell})$ which is different from $(\lambda_{1},\dots,\lambda_{\ell})$ does not contribute to the sum of Eq. 18. This completes the proof of the theorem. ∎

Remark 4.

3.3 can be proved independently from 3.2 using the explicit expression of the Pieri coefficients $\widetilde{\eta}_{\lambda,\xi}$ . We prefer here the proof based on 3.2 since it is less computational and it allows to shed some light on the properties of the operator $\Gamma$ .

3.4. Proof of Theorems 1.2 and 1.1

In this subsection we deduce Theorems 1.2 and 1.1 from Theorems 3.2 and 3.3 respectively.

Consider the transformation $\phi$ on $\Lambda$ defined by

f=\sum_{\mu}d^{f}_{\mu}(q,t)p_{\mu}[X]\longmapsto\phi(f):=\sum_{\mu}d^{f}_{\mu}(q,1/t)p_{\mu}\left[\frac{X}{1-1/t}\right],

where $d^{f}_{\mu}$ are the coefficients of $f$ in the power-sum basis. Notice that $\phi$ is invertible and

\phi^{-1}(f)=\sum_{\mu}d^{f}_{\mu}(q,1/t)p_{\mu}\left[X(1-t)\right]\quad\text{for any $f$.}

With this definition, one has

\widetilde{H}^{(q,t)}_{\lambda}=t^{n(\lambda)}\phi({J}^{(q,t)}_{\lambda}).

If $\mathcal{O}$ is an operator on modified Macdonald polynomials, then we define its integral $\bm{\mathcal{O}}$ version by the composition

(21)

\bm{\mathcal{O}}:=\phi^{-1}\cdot\mathcal{O}\cdot\phi.

In particular,

\bm{\nabla}=\phi^{-1}\cdot\nabla\cdot\phi\quad\text{ and }\quad\bm{\Delta}_{v}=\phi^{-1}\cdot\Delta_{v}\cdot\phi.

Lemma 3.6.

For every $i\geq 0$ , we have

\phi^{-1}\cdot\mathcal{P}_{\frac{ut^{i}}{1-q}}\cdot\phi=\mathcal{P}_{\frac{t^{-i}(1-t)u}{1-q}}.

Proof.

Fix a symmetric function $F[X]=\sum_{\lambda}c_{\lambda}(q,t)p_{\lambda}$ , where $c_{\lambda}(q,t)$ are the coefficients of the expansion of $F[X]$ in the power-sum basis and set $\widetilde{F}[X]:=\sum_{\lambda}c_{\lambda}(q,1/t)p_{\lambda}$ . Then

	$\displaystyle\phi\cdot\mathcal{P}_{\frac{t^{-i}(1-t)u}{1-q}}\ F[X]=\phi\operatorname{Exp}\left[\frac{t^{-i}(1-t)u}{1-q}X\right]F[X]$
	$\displaystyle=\phi\operatorname{Exp}\left[\frac{t^{-i}u}{1-q}X\right]\operatorname{Exp}\left[-\frac{t^{-i+1}u}{1-q}X\right]F[X]$
	$\displaystyle=\phi\sum_{k\geq 0}u^{k}t^{-ik}h_{k}\left[\frac{X}{1-q}\right]\sum_{k\geq 0}u^{k}t^{-(i-1)k}h_{k}\left[-\frac{X}{1-q}\right]F[X]$
	$\displaystyle=\sum_{k\geq 0}u^{k}t^{ik}h_{k}\left[\frac{X}{(1-q)(1-t^{-1})}\right]\sum_{k\geq 0}u^{k}t^{(i-1)k}h_{k}\left[-\frac{X}{(1-q)(1-t^{-1})}\right]\widetilde{F}\left[\frac{X}{1-t^{-1}}\right]$
	$\displaystyle=\operatorname{Exp}\left[\frac{t^{i}u}{1-q}\frac{X}{1-t^{-1}}\right]\operatorname{Exp}\left[-\frac{t^{i-1}u}{1-q}\frac{X}{1-t^{-1}}\right]\widetilde{F}\left[\frac{X}{1-t^{-1}}\right]$
	$\displaystyle=\operatorname{Exp}\left[\frac{u}{1-q}\frac{(t^{i}-t^{i-1})}{1-t^{-1}}X\right]\widetilde{F}\left[\frac{X}{1-t^{-1}}\right]$
	$\displaystyle=\operatorname{Exp}\left[\frac{ut^{i}}{1-q}X\right]\widetilde{F}\left[\frac{X}{1-t^{-1}}\right]=\mathcal{P}_{\frac{ut^{i}}{1-q}}\cdot\phi\ F[X]$

completing the proof. ∎

From the previous lemma and the observations above, we deduce that

(22)

\phi^{-1}\cdot\Gamma(t^{i}u,v)\cdot\phi=\bm{\Gamma}(t^{-i}u,v)\quad\text{ for every }i\geq 0.

On the other hand, we have the following lemma.

Lemma 3.7.

The following holds

\phi^{-1}\cdot\mathcal{T}_{Z}\cdot\phi=\mathcal{T}_{\frac{Z}{1-t}}.

Proof.

Both sides of the equation are clearly homomorphisms of $\mathbb{Q}(q,t)$ -algebras, hence it is enough to check the identity on the generators $p_{k}[X]$ : we have

	$\displaystyle\phi^{-1}\cdot\mathcal{T}_{Z}\cdot\phi\,p_{k}[X]$	$\displaystyle=\phi^{-1}\cdot\mathcal{T}_{Z}\,p_{k}\left[\frac{X}{1-t^{-1}}\right]=\phi^{-1}p_{k}\left[\frac{X+Z}{1-t^{-1}}\right]$
		$\displaystyle=\phi^{-1}(1-t^{-k})^{-1}(p_{k}[X]+p_{k}[Z])$
		$\displaystyle=(1-t^{k})^{-1}(p_{k}[X(1-t)]+p_{k}[Z])$
		$\displaystyle=p_{k}\left[\frac{X(1-t)}{1-t}\right]+p_{k}\left[\frac{Z}{1-t}\right]$
		$\displaystyle=p_{k}\left[X+\frac{Z}{1-t}\right]=\mathcal{T}_{\frac{Z}{1-t}}\,p_{k}[X].$

∎

Using this lemma and the remarks above, we deduce Eq. 4 by applying $\phi^{-1}$ on Eq. 14. In a similar way, we obtain Eq. 3 from Eq. 14, and the following equation from Eq. 13.

(23)		$\displaystyle\bm{\Gamma}(u,v_{1})\bm{\Gamma}(t^{-1}u,v_{2})$	$\displaystyle\cdots\bm{\Gamma}(t^{k-1}u,v_{k})\cdot 1$
	$\displaystyle=$	$\displaystyle\bm{\nabla}\mathcal{P}_{\frac{-tu}{1-q}}\bm{\nabla}^{-1}\operatorname{Exp}\left[\frac{-utX}{1-q}-\frac{u(1-t)X}{1-q}\sum_{1\leq i\leq k}t^{1-i}(1-v_{i})\right]$
	$\displaystyle=$	$\displaystyle\bm{\nabla}\mathcal{P}_{\frac{-tu}{1-q}}\bm{\nabla}^{-1}\operatorname{Exp}\left[\frac{-ut^{k-1}X}{1-q}-\frac{u(1-t)X}{1-q}\sum_{1\leq i\leq k}t^{1-i}v_{i}\right].$

4. Macdonald characters

4.1. Shifted symmetric Macdonald polynomials

We denote by $\Lambda^{*}$ the algebra of shifted symmetric functions; see 1.3. If $f$ is a shifted symmetric function, we consider its evaluation on a Young diagram $\lambda$ defined by

f(\lambda):=f(q^{\lambda_{1}},q^{\lambda_{2}},\dots,q^{\lambda_{k}},1,1,\dots).

It is well known that the space of shifted symmetric functions can be identified to a subspace of functions on Young diagrams through the map

f\longmapsto(f(\lambda))_{\lambda\in\mathbb{Y}}.

Okounkov introduced a shifted-symmetric generalization of Macdonald polynomials (see also [Kno97, Sah96]).

Theorem 4.1 (Shifted Macdonald polynomials).

[Oko98] Let $\mu$ be a partition. There exists a unique function ${J}^{*}_{\mu}(v_{1},v_{2},\dots)$ such that

(1)

${J}^{*}_{\mu}$ is shifted symmetric of degree $|\mu|$ .
(2)

(normalization property)

${J}^{*}_{\mu}(\mu)=(-1)^{|\mu|}q^{n(\mu^{\prime})}t^{-2n(\mu)}{j}^{(q,t)}_{\mu}.$
(3)

(vanishing property) for any partition $\mu\not\subset\lambda$

${J}^{*}_{\mu}(\lambda)=0.$

Moreover, the top homogeneous part of $J^{*}_{\mu}$ is ${J}^{(q,t)}_{\mu}(v_{1},t^{-1}v_{2},t^{-2}v_{3},\dots)$ .

Since these polynomials are defined by their zeros, sometimes they are referred to as interpolation polynomials.

As Macdonald polynomials form a basis of $\Lambda$ , using a triangularity argument we deduce that shifted Macdonald polynomials form a basis of $\Lambda^{*}$ . As a consequence we can linearly extend the map $J^{(q,t)}_{\mu}\longmapsto J^{*}_{\mu}$ into an isomorphism

(24)

\begin{array}[]{ccc}\Lambda&\longrightarrow&\Lambda^{*}\\ f&\longmapsto&f^{*}.\end{array}

Remark 5.

Note that it follows from linearity that if $f$ is a homogeneous symmetric function, then the top homogeneous part of $f^{*}$ is equal to $f(v_{1},t^{-1}v_{2},\dots)$ .

4.2. An explicit isomorphism between the spaces of symmetric and shifted-symmetric functions

The main purpose of this subsection is to give two explicit formulas for the isomorphism Eq. 24. The first one, Eq. 25, gives the image of a function $f^{*}$ as a shifted symmetric function while the second formula, Eq. 26, gives this image as a function on Young diagrams. The proof is based on Eq. 4. We start with the following lemma.

Lemma 4.2.

For any symmetric function $f$ , and any partition $\lambda=[\lambda_{1},\lambda_{2},\dots,\lambda_{k}]$ one has

\left\langle f,\bm{\Gamma}(1,q^{\lambda_{1}})\bm{\Gamma}(t^{-1},q^{\lambda_{2}})\cdots\bm{\Gamma}(t^{-(k-1)},q^{\lambda_{k}})\cdot 1\right\rangle_{q,t}=\left\langle\mathcal{P}_{\frac{1}{1-q}}\bm{\nabla}\cdot f,t^{-n(\lambda)}{J}^{(q,t)}_{\lambda}\right\rangle_{q,t}.

Proof.

From Eq. 4, we know that for any function $f$

\left\langle f[X],\bm{\Gamma}(u,q^{\lambda_{1}})\bm{\Gamma}(t^{-1}u,q^{\lambda_{2}})\cdots\bm{\Gamma}(t^{-(k-1)},q^{\lambda_{k}})\cdot 1\right\rangle_{q,t}=\left\langle f[X],t^{-n(\lambda)}\bm{\nabla}\mathcal{T}_{\frac{1}{u(1-t)}}{J}^{(q,t)}_{\lambda}[uX]\right\rangle_{q,t}.

Since the operator $\bm{\nabla}$ acts diagonally on the basis of Macdonald polynomials $({J}^{(q,t)}_{\lambda})_{\lambda\in\mathbb{Y}}$ and this basis is orthogonal with respect to the scalar product $\langle-,-\rangle_{q,t}$ , the operator $\bm{\nabla}$ is self dual with this scalar product. Moreover, the dual of $\mathcal{T}_{\frac{1}{u(1-t)}}$ is $\mathcal{P}_{\frac{1}{u(1-q)}}$ . We conclude by specializing $u=1$ . ∎

We have the following theorem.

Theorem 4.3.

For any symmetric function $f$ and any $k\geq 1$ , we have

(25)

f^{*}(v_{1},\dots,v_{k})=\left\langle f,\bm{\Gamma}(1,v_{1})\bm{\Gamma}(t^{-1},v_{2})\cdots\bm{\Gamma}(t^{-(k-1)},v_{k})\cdot 1\right\rangle_{q,t}.

Equivalently, for any Young diagram $\lambda$ ,

(26)

f^{*}(\lambda)=\left\langle\mathcal{P}_{\frac{1}{1-q}}\bm{\nabla}\cdot f,t^{-n(\lambda)}{J}^{(q,t)}_{\lambda}\right\rangle_{q,t}.

Proof.

First, notice that Eq. 25 implies Eq. 26 by 4.2. By definition of the isomorphism $f\mapsto f^{*}$ , we should prove that for any partition $\lambda$ the function

(27)

\left\langle J_{\lambda}^{(q,t)},\bm{\Gamma}(1,v_{1})\bm{\Gamma}(t^{-1},v_{2})\cdots\bm{\Gamma}(t^{-(k-1)},v_{k})1\right\rangle_{q,t}

satisfies the three properties of 4.1. First, from Eq. 23, we know that for any $k$

\bm{\Gamma}(u,v_{1})\cdots\bm{\Gamma}(ut^{-(k-1)},v_{k})\bm{\Gamma}(ut^{-k},1)\cdot 1=\bm{\Gamma}(u,v_{1})\cdots\bm{\Gamma}(ut^{-(k-1)},v_{k})\cdot 1.

and that for any $n,k\geq 0$ the coefficient

[u^{n}]\bm{\Gamma}(u,v_{1})\cdots\bm{\Gamma}(ut^{-(k-1)},v_{k})\cdot 1

is a homogeneous symmetric function of degree $n$ in the variables $(x_{i})_{i\geq 1}$ , with coefficients which are shifted symmetric in $(v_{i})_{1\leq i\leq k}.$ This implies that for any symmetric function $f$ , the right-hand side of Eq. 25 is a well defined shifted symmetric function in the variables $(v_{i})_{1\leq i\leq k}$ . All this gives property $(1)$ .

In order to obtain property $(2)$ , we use 4.2 with $f={J}^{(q,t)}_{\lambda}$ . We get that

	$\displaystyle\left\langle{J}^{(q,t)}_{\lambda},\bm{\Gamma}(1,q^{\lambda_{1}})\bm{\Gamma}(t^{-1},q^{\lambda_{2}})\cdots\bm{\Gamma}(t^{-(k-1)},q^{\lambda_{k}})\cdot 1\right\rangle_{q,t}$	$\displaystyle=\left\langle\mathcal{P}_{\frac{1}{1-q}}\bm{\nabla}\cdot{J}^{(q,t)}_{\lambda},t^{-n(\lambda)}{J}^{(q,t)}_{\lambda}\right\rangle_{q,t}$
		$\displaystyle=(-1)^{\|\lambda\|}q^{n(\lambda^{\prime})}t^{-2n(\lambda)}{j}^{(q,t)}_{\lambda}.$

Here we used Eq. 2, and the fact that $[z^{0}]\mathcal{P}_{\frac{z}{1-q}}=1$ . This corresponds to property $(2)$ . Finally, for any partitions $\mu$ and $\lambda$ , one has

\displaystyle\left\langle{J}^{(q,t)}_{\mu},\bm{\Gamma}(1,q^{\lambda_{1}})\bm{\Gamma}(t^{-1},q^{\lambda_{2}})\cdots\bm{\Gamma}(t^{-(k-1)},q^{\lambda_{k}})\cdot 1\right\rangle_{q,t}

\displaystyle=\left\langle\mathcal{P}_{\frac{1}{1-q}}\bm{\nabla}\cdot{J}^{(q,t)}_{\mu},t^{-n(\lambda)}{J}^{(q,t)}_{\lambda}\right\rangle_{q,t}.

But from 2.3, we know that the coefficient of ${J}^{(q,t)}_{\lambda}$ in $\mathcal{P}_{\frac{1}{1-q}}\bm{\nabla}\cdot{J}^{(q,t)}_{\mu}$ is zero unless $\mu\subset\lambda$ . This proves that (27) satisfies property $(3)$ , completing the proof of the theorem. ∎

Remark 6.

The isomorphism given in Eq. 26 has been implicitly described by Lassalle, see [Las98, Definition 1]. However, the formula of Eq. 25 seems to be new. Note that these two formulas are complementary since Eq. 25 gives the shifted symmetry property while Eq. 26 is more suitable to prove the vanishing conditions.

4.3. Macdonald characters are shifted symmetric

Recall the definition from 1.5 of Macdonald characters, i.e.

\displaystyle\bm{\widetilde{\theta}}^{(q,t)}_{\mu,k}(v_{1},v_{2},\dots):=\left\langle p_{\mu},\bm{\Gamma}(1,v_{1})\bm{\Gamma}(t^{-1},v_{2})\cdots\bm{\Gamma}(t^{-{k-1}},v_{k})\cdot 1\right\rangle_{q,t}.

We are now ready to prove 1.5.

Proof of 1.5.

By definition, for any $k\geq 1$

	$\displaystyle\bm{\widetilde{\theta}}^{(q,t)}_{\mu,k}(v_{1},v_{2},\dots,v_{k})$	$\displaystyle=\left\langle p_{\mu},\bm{\Gamma}(1,v_{1})\bm{\Gamma}(t^{-1},v_{2})\cdots\bm{\Gamma}(t^{-{k-1}},v_{k})\cdot 1\right\rangle_{q,t}$
		$\displaystyle=p_{\mu}^{*}(v_{1},v_{2},\dots,v_{k}).$

In particular, $(\bm{\widetilde{\theta}}^{(q,t)}_{\mu,k})_{k\geq 1}$ defines a shifted symmetric function. ∎

We deduce the following corollary.

Corollary 4.4.

The Macdonald characters $(\bm{\widetilde{\theta}}^{(q,t)}_{\mu})_{\mu\in\mathbb{Y}}$ form a basis of $\Lambda^{*}$ .

Proof.

From the definition of $\bm{\widetilde{\theta}}^{(q,t)}_{\mu}$ and 4.3 we have

(28)

\displaystyle\bm{\widetilde{\theta}}^{(q,t)}_{\mu}=p_{\mu}^{*}.

We conclude using the fact $(p_{\mu})_{\mu\in\mathbb{Y}}$ is basis of $\Lambda$ and that $f\longmapsto f^{*}$ is an isomorphism between $\Lambda$ and $\Lambda^{*}$ . ∎

From Eq. 26, we get that for any Young diagram $\lambda$

(29)

\bm{\widetilde{\theta}}^{(q,t)}_{\mu}(\lambda)=\left\langle\mathcal{P}_{\frac{1}{1-q}}\bm{\nabla}\cdot p_{\mu},t^{-n(\lambda)}{J}^{(q,t)}_{\lambda}\right\rangle_{q,t}.

This can be rewritten as

\bm{\widetilde{\theta}}^{(q,t)}_{\mu}(\lambda)=\left\langle p_{\mu},t^{-n(\lambda)}\bm{\nabla}\mathcal{T}_{\frac{1}{1-q}}\cdot{J}^{(q,t)}_{\lambda}\right\rangle_{q,t}.

Hence,

(30)

\bm{\widetilde{\theta}}^{(q,t)}_{\mu}(q^{\lambda_{1}},\dots q^{\lambda_{k}},1,\dots)=\left\{\begin{array}[]{cc}\left\langle p_{\mu},\bm{\nabla}h^{\perp}_{|\lambda|-|\mu|}\left[\frac{X}{1-t}\right]\cdot t^{-n(\lambda)}J^{(q,t)}_{\lambda}\right\rangle_{q,t}&\text{ if }|\mu|\leq|\lambda|\\ 0&\text{ otherwise.}\end{array}\right.

In particular,, when $|\mu|=|\lambda|$ the characters $\bm{\widetilde{\theta}}^{(q,t)}_{\mu}(\lambda)$ are given by the power-sum expansion of ${J}^{(q,t)}_{\lambda}$ :

(31)

(-1)^{|\lambda|}q^{n(\lambda^{\prime})}t^{-2n(\lambda)}{J}^{(q,t)}_{\lambda}=\sum_{\mu\vdash|\lambda|}\frac{\bm{\widetilde{\theta}}^{(q,t)}_{\mu}(\lambda)}{z_{\mu}(q,t)}p_{\mu}.

4.4. Characterization theorem

We give here a characterization theorem for $\bm{\widetilde{\theta}}^{(q,t)}_{\mu}$ . This characterization has been observed by Féray in the case of Jack polynomials and proved very useful in practice (see [BDD23]). It can be seen as an analog of 4.1 for characters.

Theorem 4.5.

Let $\mu$ be a partition. $\bm{\widetilde{\theta}}^{(q,t)}_{\mu}$ is the unique function which satisfies the following properties.

(1)

$\bm{\widetilde{\theta}}^{(q,t)}_{\mu}$ is shifted symmetric of degree $|\mu|$ .
(2)

$\bm{\widetilde{\theta}}^{(q,t)}_{\mu}(\lambda)=0$ for any partition $|\lambda|<|\mu|$ .
(3)

the top homogeneous part of $\bm{\widetilde{\theta}}^{(q,t)}_{\mu}$ is $p_{\mu}(v_{1},t^{-1}v_{2},t^{-2}v_{3},\dots).$

The proof is very similar. We start by the following lemma.

Lemma 4.6.

Let $n\geq 1$ . If $G$ is shifted symmetric function of degree less or equal to $n$ with

(32)

G(\lambda)=0\text{ for $|\lambda|\leq n$}.

Then $G=0$ .

Proof.

We expand $G$ in the $J^{*}_{\xi}$ basis

(33)

G=\sum_{\xi}c_{\xi}J^{*}_{\xi}.

As $\deg(G)\leq n,$ the sum can be restricted to partitions $\xi$ of size at most $n$ . We will prove by contradiction that $G=0$ , i.e. that $c_{\xi}=0$ for all partitions $\xi$ with $|\xi|\leq n$ . Assume this is not the case and consider a partition $\xi_{0}$ of minimal size such that $c_{\xi_{0}}\neq 0$ . Eq. 32 gives $G(\xi_{0})=0$ since $|\xi_{0}|\leq n$ . On the other hand, $J_{\xi}^{*}(\xi_{0})=0$ if $\xi_{0}$ does not contain $\xi$ (see property (3) of 4.1). Therefore the right hand side of Eq. 33 evaluated on $\xi_{0}$ vanishes for all partitions $\xi$ except for $\xi=\xi_{0}$ . Moreover, $c_{\xi_{0}}\neq 0$ by the assumptions and $J^{*}_{\xi_{0}}(\xi_{0})\neq 0$ from property (2) of 4.1. Therefore $G(\xi_{0})=c_{\xi_{0}}J^{*}_{\xi_{0}}(\xi_{0})\neq 0$ , and we have reached a contradiction. Hence, $G=0$ as required. ∎

We now prove the characterization theorem.

Proof of 4.5.

We start by proving that $\bm{\widetilde{\theta}}^{(q,t)}_{\mu}$ satisfies these three properties. The first property is given in 1.5, and from 5, we know that its top homogeneous part is $p_{\mu}(v_{1},t^{-1}v_{2},t^{-2}v_{3},\dots)$ . Moreover, $\bm{\widetilde{\theta}}^{(q,t)}_{\mu}$ is a linear combination of $J_{\xi}^{*}$ for $\xi$ of size $|\mu|$ . This gives the vanishing property.

Let us now prove the uniqueness. Let $F$ be a shifted symmetric function of degree $|\mu|$ with the same top degree part as $\bm{\widetilde{\theta}}^{(q,t)}_{\mu}$ , and such that $F(\lambda)=0$ for any $|\lambda|<|\mu|$ . Set $G:=F-\bm{\widetilde{\theta}}^{(q,t)}_{\mu}$ . Then $G$ is a shifted symmetric function of degree at most $|\mu|-1$ with $G(\lambda)=0\text{ for $|\lambda|<|\mu|$}.$ Using 4.6 we get that $G=0$ hence $F=\bm{\widetilde{\theta}}^{(q,t)}_{\mu}$ which gives the uniqueness. ∎

4.5. Positivity conjectures about the characters $\bm{\widetilde{\theta}}^{(q,t)}_{\mu}$

We conclude this section with some intriguing positivity conjectures about the operator $\bm{\Gamma}$ .

Conjecture 1.

The operator $\bm{\Gamma}(z,v)$ acts positively on the basis $p_{\mu}\left[X\frac{1-t}{1-q}\right]$ in the variables $q^{\prime},\gamma,-v,-z$ . More precisely, if $\mu$ and $\nu$ are two partitions such that $|\nu|-|\mu|=n$ , then

(-1)^{n}t^{|\mu|}\left\langle[z^{n}]\bm{\Gamma}(z,v)\cdot p_{\mu}\left[X\frac{1-t}{1-q}\right],p_{\nu}\left[X\frac{1-q}{1-t}\right]\right\rangle

is a polynomial in the variables $-v,q^{\prime},\gamma$ with non-negative integer coefficients.

This conjecture has been tested for $|\nu|\leq 9$ . We have the following consequence of 1 which is closely related to a conjecture about Macdonald characters we formulate in the next section; see 4.

Proposition 4.7.

1 implies that $(-1)^{|\mu|}t^{2(k-1)|\mu|}\bm{\widetilde{\theta}}^{(q,t)}_{\mu}(v_{1},v_{2},\dots,v_{k})$ is a polynomial in the variables $-v_{1},\cdots,-v_{k}$ and the parameters $q^{\prime}$ and $\gamma$ with non-negative coefficients.

Proof.

Let us assume that 1 holds. We want to prove by induction on $k$ that for any $n$

(-1)^{n}t^{2(k-1)n}[z^{n}]\bm{\Gamma}(z,v_{1})\bm{\Gamma}(t^{-1}z,v_{2})\cdots\bm{\Gamma}(t^{-(k-1)}z,v_{k})\cdot 1

has a positive polynomial expansion on the basis $\left(p_{\mu}\left[X\frac{1-t}{1-q}\right]\right)_{\mu\vdash n}.$ This would imply the claim of the proposition by the definition of Macdonald characters Eq. 5. For $k=1$ this is a direct consequence of 1. We assume now that the induction assumption holds for $k$ . We then get that

(-1)^{n}t^{2(k-1)n}[z^{n}]\bm{\Gamma}(z,v_{2})\bm{\Gamma}(t^{-1}z,v_{3})\cdots\bm{\Gamma}(t^{-(k-1)}z,v_{k+1})\cdot 1\\ =(-1)^{n}t^{(2k-1)n}[z^{n}]\bm{\Gamma}(t^{-1}z,v_{2})\bm{\Gamma}(t^{-2}z,v_{3})\cdots\bm{\Gamma}(t^{-k}z,v_{k+1})\cdot 1

also has positive expansion. Applying $t^{n}\bm{\Gamma}(z,v_{1})$ on the left and using 1, we obtain that

\left((-1)^{m}[z^{m}]\bm{\Gamma}(z,v_{1})\right)(-1)^{n}t^{2kn}[z^{n}]\bm{\Gamma}(t^{-1}z,v_{2})\bm{\Gamma}(t^{-2}z,v_{3})\cdots\bm{\Gamma}(t^{-k}z,v_{k+1})\cdot 1,

has positive expansion on $\left(p_{\mu}\left[X\frac{1-t}{1-q}\right]\right)_{\mu\vdash(n+m)}.$ In particular this is also the case for

(-1)^{m+n}t^{2k(n+m)}\left([z^{m}]\bm{\Gamma}(z,v_{1})\right)[z^{n}]\bm{\Gamma}(t^{-1}z,v_{2})\bm{\Gamma}(t^{-2}z,v_{3})\cdots\bm{\Gamma}(t^{-k}z,v_{k+1})\cdot 1.

Since this holds true for any $n,m\geq 0$ , we deduce the induction hypothesis for $k+1$ . ∎

5. Macdonald versions for some Jack conjectures

Jack polynomials are symmetric functions which depend on a deformation parameter $\alpha$ . We briefly present here some of the most important results and conjectures related to Jack polynomials. We then introduce a new change of variables which allows us to generalize these conjectures to Macdonald polynomials.

The section is organized as follows. In Section 5.1, we recall the definition of Jack polynomials and we introduce a new parametrization of Macdonald polynomials which is directly related to Jack polynomials. We then discuss Macdonald generalizations of Stanley’s and Lassalle’s conjectures in Sections 5.2 and 5.3 respectively. The rest of the subsections are dedicated to discuss a generalization of Goulden–Jackson’s Matchings-Jack and $b$ -conjectures. In Section 5.4 we state the generalized conjectures. We then discuss in Section 5.5 the connection of the generalized Matching-Jack conjecture to the structure coefficients of Macdonald characters and we give a reformulation of this conjecture in Section 5.6 with the super nabla operator. We finally discuss some special cases of these conjectures in Section 5.7.

5.1. Jack polynomials and a new normalization of Macdonald polynomials

Jack polynomials can be obtained from the integral form of Macdonald polynomials as follows (see [Mac95, Chapter VI, eq (10.23)])

(34)

\lim_{t\rightarrow 1}\frac{J_{\lambda}^{(q=1+\alpha(t-1),t)}}{(1-t)^{|\lambda|}}=J^{(\alpha)}_{\lambda}.

We denote the $\langle-,-\rangle_{\alpha}$ , the scalar product defined on power-sum functions by

\langle p_{\mu},p_{\nu}\rangle_{\alpha}=\delta_{\mu,\nu}z_{\mu}\alpha^{\ell(\mu)}.

Jack polynomials are orthogonal with respect to this scalar product. We denote by $j^{(\alpha)}_{\lambda}$ their squared norm, i.e.

\langle J^{(\alpha)}_{\lambda},J_{\mu}^{(\alpha)}\rangle_{\alpha}=\delta_{\lambda,\mu}j^{(\alpha)}_{\lambda}.

We consider the following normalization of Macdonald polynomials

{\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}:=\frac{{J}^{(q,t)}_{\lambda}}{(1-t)^{|\lambda|}}_{\big{|}q=1+\gamma\alpha,t=1+\gamma}.

In the following, the parameters $(\alpha,\gamma)$ will be always related to $(q,t)$ by

(35)

\left\{\begin{array}[]{ll}q=1+\gamma\alpha\\ t=1+\gamma\end{array}\right.\longleftrightarrow\left\{\begin{array}[]{ll}\alpha=\frac{1-q}{1-t}\\ \gamma=t-1.\end{array}\right.

Note that from Eq. 34, we get

(36)

\mathfrak{J}^{(\alpha,\gamma=0)}_{\lambda}=J^{(\alpha)}_{\lambda}.

Unlike the functions ${J}^{(q,t)}_{\lambda}$ , the normalized functions ${\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}$ are positive in the monomial basis.

Proposition 5.1.

The coefficient of ${\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}$ in the monomial basis are polynomial in $\alpha$ and $\gamma$ with non-negative integer coefficients.

Proof.

This can be easily obtained from the combinatorial interpretation given in [HHL05, Proposition 8.1] for ${J}^{(q,t)}_{\lambda}$ . ∎

Our new $(\alpha,\gamma)$ -reparametrization of Macdonald polynomials will allow us to formulate Macdonald generalizations of Stanley, Lassalle and Goulden–Jackson’s conjectures. In the following, we recall these conjectures in the Jack case and we then state their Macdonald generalizations.

5.2. Stanley’s conjecture

5.2.1. Jack case

In his seminal work [Sta89], Stanley formulated the following positivity conjecture about the structure coefficients of Jack polynomials (see [Sta89, Conjecture 8.5]).

Conjecture 2.

[Sta89] For arbitrary partitions $\lambda$ , $\mu$ and $\nu$ ,

\langle J^{(\alpha)}_{\lambda}J^{(\alpha)}_{\mu},J^{(\alpha)}_{\nu}\rangle_{\alpha}

is a polynomial in $\alpha$ with non-negative integer coefficients.

This conjecture is wide open, and an analog for Shifted Jack polynomials has been proposed in [AF19].

5.2.2. Macdonald generalization

Conjecture 3 (Macdonald version of Stanley’s conjecture).

For any partitions $\lambda,\mu,\nu$ the quantity

\langle{\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}{\mathfrak{J}}^{(\alpha,\gamma)}_{\mu},{\mathfrak{J}}^{(\alpha,\gamma)}_{\nu}\rangle_{q,t}

is a polynomial in the parameters $\alpha$ and $\gamma$ with non-negative integer coefficients.

This conjecture has been tested for $|\nu|\leq 9$ . Stanley’s conjecture corresponds to the case $\gamma=0$ of 3.

5.3. Lassalle’s conjecture

5.3.1. Jack case

Jack characters have been introduced by Lassalle [Las08] as a one parameter deformation of the characters of the symmetric group.

Definition 5.2 (Jack characters).

Fix a partition $\mu$ . The Jack character $\theta^{(\alpha)}_{\mu}$ is the function on Young diagrams $\lambda$ defined by

\theta^{(\alpha)}_{\mu}(\lambda):=\left\{\begin{array}[]{ll}\left[p_{\mu}\right]\frac{1}{(|\lambda|-|\mu|)!}(p_{1}^{\perp})^{|\lambda|-|\mu|}J^{(\alpha)}_{\lambda}&\mbox{ if $|\mu|\leq|\lambda|$}\\ 0&\mbox{ if }|\lambda|<|\mu|.\end{array}\right.

Lassalle’s conjecture, formulated in [Las08] and proved in [BDD23], suggests that the character $\theta^{(\alpha)}_{\mu}(\lambda)$ is a positive polynomial in $b:=\alpha-1$ , and some coordinates of $\lambda$ called multirectangular coordinates. We are here interested in a weak version of this result, which we generalize to the Macdonald case.

Theorem 5.3 ([BDD23]).

Fix a partition $\mu$ . The normalized Jack character $(-1)^{|\mu|}z_{\mu}\theta^{(\alpha)}_{\mu}(\lambda)$ is a polynomial in the variables $b:=\alpha-1,-\alpha\lambda_{1},-\alpha\lambda_{2}\dots$ with non-negative integer coefficients.

5.3.2. Macdonald generalization

We start by introducing a normalization of Macdonald characters which is directly related to Jack characters.

(37)

{\bm{\theta}}^{(\alpha,\gamma)}_{\mu}(s_{1},s_{2},\dots):=\frac{1}{\gamma^{|\mu|}z_{\mu}^{(q,t)}}\bm{\widetilde{\theta}}^{(q,t)}_{\mu}(1+\alpha\gamma s_{1},1+\alpha\gamma s_{2},\dots).

Note that ${\bm{\theta}}^{(\alpha,\gamma)}_{\mu}$ is symmetric in the variables $t^{-i}s_{i}+\frac{t^{-i}}{\alpha\gamma}$ . For any partition $\lambda$ , we denote

(38)

{\bm{\theta}}^{(\alpha,\gamma)}_{\mu}(\lambda):={\bm{\theta}}^{(\alpha,\gamma)}_{\mu}\left(\frac{q^{\lambda_{1}}-1}{q-1},\frac{q^{\lambda_{2}}-1}{q-1},\dots\right)=\frac{\bm{\widetilde{\theta}}^{(q,t)}_{\mu}(\lambda)}{\gamma^{|\mu|}z_{\mu}(q,t)}.

Hence,

(39)

{\bm{\theta}}^{(\alpha,\gamma)}_{\mu}(\lambda)=\left\{\begin{array}[]{cc}[p_{\mu}](-1)^{|\mu|}\bm{\nabla}(1-t)^{|\lambda|-|\mu|}h^{\perp}_{|\lambda|-|\mu|}\left[\frac{X}{1-t}\right]\cdot t^{-n(\lambda)}{\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}&\text{ if }|\mu|\leq|\lambda|\\ 0&\text{ otherwise.}\end{array}\right.

In particular, when $|\lambda|=|\mu|$ the characters ${\bm{\theta}}^{(\alpha,\gamma)}_{\mu}(\lambda)$ are given by the expansion

(40)

q^{n(\lambda^{\prime})}t^{-2n(\lambda)}{\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}=\sum_{\mu\vdash|\lambda|}{\bm{\theta}}^{(\alpha,\gamma)}_{\mu}(\lambda)p_{\mu}.

The Jack normalization of Macdonald characters are related to Jack characters by the following proposition.

Proposition 5.4.

For any partitions $\mu$ and $\lambda$ , we have

\lim_{\gamma\mapsto 0}{\bm{\theta}}^{(\alpha,\gamma)}_{\mu}(\lambda)=\theta^{(\alpha)}_{\mu}(\lambda).

Proof.

Since $t\rightarrow 1$ as $\gamma\mapsto 0$ , and the operator $\bm{\nabla}$ acts on a homogeneous functions of degree $n$ as a multiplication by $(-1)^{n}$ . Moreover,

\lim_{t\rightarrow 1}h_{r}^{\perp}\left[\frac{X}{1-t}\right](1-t)^{r}=\lim_{t\rightarrow 1}(1-t)^{r}\sum_{\nu\vdash r}\prod_{i\in\nu}\frac{p_{\nu_{i}}^{\perp}}{(1-t^{\nu_{i}})z_{\nu}}=\frac{(p_{1}^{\perp})^{r}}{r!}.

Eq. 39 allows to conclude.∎

Remark 7.

Actually, $\bm{\theta}_{\mu}^{(\alpha,\gamma=0)}$ coincides also as a polynomial in the variables $(s_{i})$ with the Jack character $\theta_{\mu}^{(\alpha)}$ . This can be shown using the previous proposition and the fact that $\lim_{\gamma\rightarrow 0}{\bm{\theta}}^{(\alpha,\gamma)}_{\mu}(s_{1},s_{2},\dots)$ is symmetric in the variables $s_{i}-i/\alpha$ .

These characters seem to satisfy the following conjecture, tested for $k\leq 3$ and $|\mu|\leq 7$ .

Conjecture 4.

Fix $k\geq 1$ and $\mu\in\mathbb{Y}$ . Then, $(-1)^{|\mu|}t^{(k-1)|\mu|}z_{\mu}(q,t){\bm{\theta}}^{(\alpha,\gamma)}_{\mu}(s_{1},s_{2},\dots,s_{k})$ is a polynomial in $\gamma,b:=\alpha-1,-\alpha s_{1},-\alpha s_{2}\dots,-\alpha s_{k}$ with non-negative integer coefficients.

5.4. Goulden and Jackson’s conjectures

5.4.1. Jack case

We start by recalling the Matching-Jack and $b$ -conjectures formulated by Goulden and Jackson in [GJ96a] for Jack polynomials. Let $Y:=y_{1}+y_{2}+\cdots$ and $Z:=z_{1}+z_{2}+\cdots$ be two additional alphabets of variables. We consider the two families of coefficients $c^{\lambda}_{\mu,\nu}(\alpha)$ and $h^{\lambda}_{\mu,\nu}(\alpha)$ indexed by partitions of the same size and defined by

\displaystyle\sum_{\lambda\in\mathbb{Y}}u^{|\lambda|}\frac{J^{(\alpha)}_{\lambda}[X]J^{(\alpha)}_{\lambda}[Y]J^{(\alpha)}_{\lambda}[Z]}{j^{(\alpha)}_{\lambda}}=\sum_{m\geq 0}\sum_{\pi,\mu,\nu\vdash m}\frac{u^{m}c^{\pi}_{\mu,\nu}(\alpha)}{z_{\pi}\alpha^{\ell(\pi)}}p_{\pi}[X]p_{\mu}[Y]p_{\nu}[Z],

\displaystyle\log\left(\sum_{\lambda\in\mathbb{Y}}u^{|\lambda|}\frac{J^{(\alpha)}_{\lambda}[X]J^{(\alpha)}_{\lambda}[Y]J^{(\alpha)}_{\lambda}[Z]}{j^{(\alpha)}_{\lambda}}\right)=\sum_{m\geq 0}\sum_{\pi,\mu,\nu\vdash m}\frac{u^{m}h^{\pi}_{\mu,\nu}(\alpha)}{\alpha m}p_{\pi}[X]p_{\mu}[Y]p_{\nu}[Z].

For $\alpha=1$ (resp. $\alpha=2$ ), the series above are known to count bipartite graphs on orientable surfaces (resp. surfaces orientable or not) called maps, see [GJ96b]. Goulden and Jackson have formulated the following two conjectures; see [GJ96a, Conjecture 3.5] and [GJ96a, Conjecture 6.2].

Conjecture 5 ( Matchings-Jack conjecture [GJ96a]).

The coefficients $c^{\pi}_{\mu,\nu}$ are polynomials in the parameter $b:=\alpha-1$ with non-negative integer coefficient.

Conjecture 6 ( $b$ -conjecture [GJ96a]).

The coefficients $h^{\pi}_{\mu,\nu}$ are polynomials in the parameter $b:=\alpha-1$ with non-negative integer coefficient.

These two conjectures have combinatorial reformulations in terms of maps counted with “non-orientability” weights. As mentioned in the introduction both of the conjectures are still open. The integrality part in the Matchings-Jack conjecture has been proved in [BD23] and will be useful in the next subsection.

Theorem 5.5.

[BD23] For any partitions $\pi,\mu,\nu\vdash n\geq 1$ , the coefficient $c^{\pi}_{\mu,\nu}$ is a polynomial in $b$ with integer coefficients.

5.4.2. Macdonald generalization

We define the coefficients $\bm{c}^{\pi}_{\mu,\nu}$ and $\bm{h}^{\pi}_{\mu,\nu}$ for partitions $\pi,\mu$ and $\nu$ of the same size by

(41)

\sum_{\lambda\in\mathbb{Y}}u^{|\lambda|}t^{-2n(\lambda)}q^{n(\lambda^{\prime})}\frac{{\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}[X]{\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}[Y]{\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}[Z]}{\widetilde{j}^{(\alpha,\gamma)}_{\lambda}}\\ =\sum_{m\geq 0}\sum_{\pi,\mu,\nu\vdash m}\frac{u^{m}\bm{c}^{\pi}_{\mu,\nu}(\alpha,\gamma)}{z_{\pi}(q,t)}p_{\pi}[X]p_{\mu}[Y]p_{\nu}[Z],

and

(42)

\log\left(\sum_{\lambda\in\mathbb{Y}}u^{|\lambda|}t^{-2n(\lambda)}q^{n(\lambda^{\prime})}\frac{{\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}[X]{\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}[Y]{\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}[Z]}{\widetilde{j}^{(\alpha,\gamma)}_{\lambda}}\right)\\ =\sum_{m\geq 0}\sum_{\pi,\mu,\nu\vdash m}\frac{u^{m}\bm{h}^{\pi}_{\mu,\nu}(\alpha,\gamma)}{\alpha[m]_{q}}p_{\pi}[X]p_{\mu}[Y]p_{\nu}[Z],

where

(43)

\widetilde{j}^{(\alpha,\gamma)}_{\lambda}:=\gamma^{-2|\lambda|}{j}^{(q,t)}_{\lambda}=\left\langle{\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda},{\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}\right\rangle_{q,t},

and

[m]_{q}:=1+q+\dots+q^{m-1}.

It is straightforward from the definitions and Eq. 36 that

\bm{c}^{\pi}_{\mu,\nu}(\alpha,\gamma=0)=c^{\pi}_{\mu,\nu}(\alpha)\text{ and }\bm{h}^{\pi}_{\mu,\nu}(\alpha,\gamma=0)=h^{\pi}_{\mu,\nu}(\alpha).

Conjecture 7 (A Macdonald generalization of the Matchings-Jack conjecture).

For any positive integer $n$ and partitions $\pi,\mu,\nu$ of $n$ , the quantity

(1+\gamma)^{n(n-1)}z_{\mu}z_{\nu}\bm{c}^{\pi}_{\mu,\nu}(\alpha,\gamma)

is a polynomial in $b$ and $\gamma$ with non-negative integer coefficients.

Conjecture 8 (A Macdonald generalization of the $b$ -conjecture).

For any positive integer $n$ and partitions $\pi,\mu,\nu$ of $n$ , the quantity

(1+\gamma)^{n(n-1)}z_{\pi}z_{\mu}z_{\nu}\bm{h}^{\pi}_{\mu,\nu}(\alpha,\gamma)

is a polynomial in $b$ and $\gamma$ with non-negative integer coefficients.

7 has been tested for $n\leq 8$ and 8 for $n\leq 9$ .

Remark 8.

In Equations 41 and Eq. 42 it seems possible to change the factor $t^{-2n(\lambda)}q^{n(\lambda^{\prime})}$ by $t^{-n(\lambda)}(t^{-n(\lambda)}q^{n(\lambda^{\prime})})^{r}$ for some $r\geq 0$ and the conjectures above still hold. However, we will prove in Section 5.5 that for the specific choice of $r=1$ , the coefficients $\bm{c}^{\pi}_{\mu,\nu}(\alpha,\gamma)$ are a special case of the structure coefficients of the characters ${\bm{\theta}}^{(\alpha,\gamma)}_{\mu}$ . This implies that in some sense these coefficients are a natural two parameters generalization of the coefficients considered by Goulden and Jackson and justifies the factor $t^{-2n(\lambda)}q^{n(\lambda^{\prime})}$ which appears in the previous definitions.

Proposition 5.6.

7 implies 5.

Proof.

7 implies that the coefficients $c^{\pi}_{\mu,\nu}(\alpha)$ are polynomials in $\alpha$ with positive coefficients. But these polynomials have integer coefficients by [BD23]. This concludes the proof. ∎

Note also that in a similar way the positivity in 8 implies the positivity in 6.

5.5. Connection with the Structure coefficients $\bm{g}^{\pi}_{\mu,\nu}$ of Macdonald characters

In this subsection, we consider the structure coefficients of Macdonald characters, and we prove that in some sense they generalize the coefficients $\bm{c}^{\pi}_{\mu,\nu}$ considered in Section 5.4.2.

Note that ${\bm{\theta}}^{(\alpha,\gamma)}_{\mu}(\lambda)$ is obtained from $\bm{\widetilde{\theta}}^{(q,t)}_{\mu}$ by a normalization by a scalar and a change of variables (see Eq. 38), hence their structure coefficients are well defined:

(44)

{\bm{\theta}}^{(\alpha,\gamma)}_{\mu}{\bm{\theta}}^{(\alpha,\gamma)}_{\nu}=\sum_{\pi}\bm{g}^{\pi}_{\mu,\nu}(\alpha,\gamma){\bm{\theta}}^{(\alpha,\gamma)}_{\pi}.

It follows from 5.4 that the coefficients $\bm{g}^{\pi}_{\mu,\nu}(\alpha,\gamma)$ are a two parameter generalization of structure coefficients of Jack characters $\theta^{(\alpha)}_{\mu}$ introduced by Dołe˛ga and Féray in [DF16] (see also [Śni19]):

\theta^{(\alpha)}_{\mu}\theta^{(\alpha)}_{\nu}=\sum_{\pi}\bm{g}^{\pi}_{\mu,\nu}(\alpha,\gamma=0)\theta^{(\alpha)}_{\pi}.

In the following, we will prove that in the case $|\pi|=|\mu|=|\nu|$ the coefficients $\bm{g}^{\pi}_{\mu,\nu}(\alpha,\gamma)$ coincide with the coefficients $\bm{c}^{\pi}_{\mu,\nu}(\alpha,\gamma)$ defined in Section 5.4.2. The proof is very similar to the one given in [DF16] for the Jack case. We start by proving some properties of the coefficients $\bm{g}^{\pi}_{\mu,\nu}$ .

Lemma 5.7.

The coefficient $\bm{g}^{\pi}_{\mu,\nu}$ is 0 unless

\max(|\mu|,|\nu|)\leq|\pi|\leq|\mu|+|\nu|.

Proof.

The upper bound is a consequence of the fact that ${\bm{\theta}}^{(\alpha,\gamma)}_{\mu}{\bm{\theta}}^{(\alpha,\gamma)}_{\nu}$ is a shifted symmetric function of degree $|\mu|+|\nu|$ and that $\left({\bm{\theta}}^{(\alpha,\gamma)}_{\pi}\right)_{|\pi|\leq d}$ is a basis of the space of shifted symmetric functions of degree less or equal than $d$ . On the other hand, for any partition $\lambda$ such that $|\lambda|<\max(|\mu|,|\nu|)$ , one has by the vanishing condition that ${\bm{\theta}}^{(\alpha,\gamma)}_{\mu}(\lambda){\bm{\theta}}^{(\alpha,\gamma)}_{\nu}(\lambda)=0$ , and also that

\sum_{|\pi|\geq\max(\mu,\nu)}\bm{g}^{\pi}_{\mu,\nu}(\alpha,\gamma){\bm{\theta}}^{(\alpha,\gamma)}_{\pi}(\lambda)=0.

Combining these two equations with Eq. 44, we get that

\sum_{|\pi|<\max(|\mu|,|\nu|)}\bm{g}^{\pi}_{\mu,\nu}(\alpha,\gamma){\bm{\theta}}^{(\alpha,\gamma)}_{\pi}(\lambda)=0,\quad\text{ for any }|\lambda|<\max(|\mu|,|\nu|).

But $\sum_{|\pi|<\max(\mu,\nu)}\bm{g}^{\pi}_{\mu,\nu}(\alpha,\gamma){\bm{\theta}}^{(\alpha,\gamma)}_{\pi}$ is a shifted symmetric function of degree smaller than $\max(|\mu|,|\nu|)$ . Using 4.6, we deduce that it is identically equal to 0, therefore $\bm{g}^{\pi}_{\mu,\nu}(\alpha,\gamma)=0$ for any $|\pi|<\max(\mu,\nu)$ . ∎

We deduce the following corollary.

Corollary 5.8.

Fix a positive integer $m$ and three partitions $\lambda,\mu,\nu\vdash m$ . Then

{\bm{\theta}}^{(\alpha,\gamma)}_{\mu}(\lambda){\bm{\theta}}^{(\alpha,\gamma)}_{\nu}(\lambda)=\sum_{\pi\vdash m}\bm{g}^{\pi}_{\mu,\nu}(\alpha,\gamma){\bm{\theta}}^{(\alpha,\gamma)}_{\pi}(\lambda).

Proof.

We start by evaluating Eq. 44 in $\lambda$ . From the vanishing condition we know that partitions $\pi$ of size larger than $m$ do not contribute to the sum. But applying 5.7 we get that $\bm{g}^{\pi}_{\mu,\nu}=0$ if $|\pi|<m$ . This concludes the proof. ∎

We then have the following proposition.

Proposition 5.9.

Let $\pi,\mu$ and $\nu$ be three partitions of the same size $m$ . Then

\bm{c}^{\pi}_{\mu,\nu}(\alpha,\gamma)=\bm{g}^{\pi}_{\mu,\nu}(\alpha,\gamma).

Proof.

We introduce for each $\mu,\nu\vdash m$ the two generating series

C_{\mu,\nu}:=\sum_{\pi\vdash m}\frac{\bm{c}^{\pi}_{\mu,\nu}(\alpha,\gamma)}{z_{\pi}(q,t)}p_{\pi}[X],

and

G_{\mu,\nu}:=\sum_{\pi\vdash m}\frac{\bm{g}^{\pi}_{\mu,\nu}(\alpha,\gamma)}{z_{\pi}(q,t)}p_{\pi}[X].

We want to prove that these two series are equal. From the definition of the coefficients $\bm{c}^{\pi}_{\mu,\nu}$ and Eq. 40 we have

\displaystyle C_{\mu,\nu}=\sum_{\lambda\vdash m}\frac{{\bm{\theta}}^{(\alpha,\gamma)}_{\mu}(\lambda){\bm{\theta}}^{(\alpha,\gamma)}_{\nu}(\lambda)}{t^{-2n(\lambda)}q^{n(\lambda^{\prime})}\widetilde{j}^{(\alpha,\gamma)}_{\lambda}}{\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}[X].

Using 5.8, we get that

(45)

C_{\mu,\nu}=\sum_{\pi,\lambda\vdash m}\frac{\bm{g}^{\pi}_{\mu,\nu}(\alpha,\gamma){\bm{\theta}}^{(\alpha,\gamma)}_{\pi}(\lambda)}{t^{-2n(\lambda)}q^{n(\lambda^{\prime})}\widetilde{j}^{(\alpha,\gamma)}_{\lambda}}{\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}[X].

On the other hand, using the fact that both Macdonald polynomials and the power-sum functions are orthogonal families, Eq. 40 can be inverted as follows

\frac{p_{\pi}}{z_{\pi}(q,t)}=\sum_{\lambda\vdash m}\frac{{\bm{\theta}}^{(\alpha,\gamma)}_{\pi}(\lambda)}{t^{-2n(\lambda)}q^{n(\lambda^{\prime})}\widetilde{j}^{(\alpha,\gamma)}_{\lambda}}{\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}.

Hence, Eq. 45 becomes

C_{\mu,\nu}=\sum_{\pi\vdash m}\frac{\bm{g}^{\pi}_{\mu,\nu}}{z_{\pi}(q,t)}p_{\pi}[X].

This is precisely the series $G_{\mu,\nu}$ , which concludes the proof of the proposition. ∎

Let $f$ be the function defined on tuples of non-negative integers $(n_{1},n_{2},k)$ by

f(n_{1},n_{2},k):=(M-m)(M+m-k)+m(m-1)-(k-M)(k-M-1),

where

M:=\max(n_{1},n_{2})\text{ and }m=\min(n_{1},n_{2}).

We consider the following conjecture.

Conjecture 9.

Let $\pi,\mu,\nu$ be three partitions. Then, the normalized coefficients

(1+\gamma)^{f(|\mu|,|\nu|,|\pi|)}z_{\mu}z_{\nu}\bm{g}^{\pi}_{\mu,\nu}

are polynomials in $b:=\alpha-1$ and $\gamma$ with non-negative integer coefficients.

This conjecture has been tested for $|\pi|,|\mu|,|\nu|\leq 7$ . Since $f(n,n,n)=n(n-1)$ , and given 5.9, it is easy to check that 7 is a special case of 9.

Remark 9.

Śniady has formulated a positivity conjecture about the structure coefficients of Jack characters [Śni19, Conjecture 2.2]. This conjecture is related to the case $\gamma=0$ in 9 but the normalizations are different.

5.6. Reformulation with the super nabla operator

The super nabla operator has been recently introduced in [BHIR23]. It is defined by its action on modified Macdonald polynomials

\nabla_{Y}\widetilde{H}_{\lambda}[X]=\widetilde{H}_{\lambda}[X]\widetilde{H}_{\lambda}[Y],

where $Y:=y_{1}+y_{2}+\dots$ is a second alphabet of variables. We consider here the integral version of this operator $\bm{\nabla}_{Y}$ defined by

\bm{\nabla}_{Y}{J}^{(q,t)}_{\lambda}[X]=t^{-n(\lambda)}{J}^{(q,t)}_{\lambda}[X]{J}^{(q,t)}_{\lambda}\left[Y\right].

Proposition 5.10.

Let $\pi\vdash m$ .

\frac{1}{\gamma^{m}}\bm{\nabla}\cdot\bm{\nabla}_{Y}\cdot p_{\pi}[X]=\sum_{\mu,\nu\vdash m}\bm{c}^{\pi}_{\mu,\nu}(\alpha,\gamma)p_{\mu}[X]p_{\nu}[Y].

Proof.

Let $Z$ be a third alphabet of variables, and let $\widetilde{\bm{\nabla}}$ and $\widetilde{\bm{\nabla}}_{Y}$ denote respectively the nabla and the super nabla operators acting on the space of symmetric functions in the alphabet $Z$ . Using the fact that power-sum functions form an orthogonal basis, one has

\displaystyle\frac{1}{\gamma^{m}}\bm{\nabla}\bm{\nabla}_{Y}\cdot p_{\pi}[X]

\displaystyle=\left\langle\frac{1}{\gamma^{m}}\widetilde{\bm{\nabla}}\widetilde{\bm{\nabla}}_{Y}\cdot p_{\pi}[Z],\sum_{\pi\vdash m}\frac{p_{\mu}[X]p_{\mu}[Z]}{z_{\mu}(q,t)}\right\rangle_{q,t},

where the scalar product is taken with respect to the alphabet $Z$ . But using the Cauchy identity Eq. 7 we get

\displaystyle\frac{1}{\gamma^{m}}\bm{\nabla}\bm{\nabla}_{Y}\cdot p_{\pi}[X]

\displaystyle=\left\langle\frac{1}{\gamma^{m}}\widetilde{\bm{\nabla}}\widetilde{\bm{\nabla}}_{Y}\cdot p_{\pi}[Z],\sum_{\lambda\vdash m}\frac{{J}^{(q,t)}_{\lambda}[X]{J}^{(q,t)}_{\lambda}[Z]}{{j}^{(q,t)}_{\lambda}}\right\rangle_{q,t}.

Using the fact that the nabla operators are self-dual, we get

	$\displaystyle\frac{1}{\gamma^{m}}\bm{\nabla}\bm{\nabla}_{Y}\cdot p_{\pi}[X]$	$\displaystyle=\left\langle\frac{1}{\gamma^{m}}p_{\pi}[Z],\widetilde{\bm{\nabla}}_{Y}\widetilde{\bm{\nabla}}\cdot\sum_{\lambda\vdash m}\frac{{J}^{(q,t)}_{\lambda}[X]{J}^{(q,t)}_{\lambda}[Z]}{{j}^{(q,t)}_{\lambda}}\right\rangle_{q,t}$
		$\displaystyle=\left\langle\frac{1}{\gamma^{m}}p_{\pi}[Z],(-1)^{m}\sum_{\lambda\vdash m}q^{n(\lambda^{\prime})}t^{-2n(\lambda)}\frac{{J}^{(q,t)}_{\lambda}[X]{J}^{(q,t)}_{\lambda}[Z]{J}^{(q,t)}_{\lambda}[Y]}{{j}^{(q,t)}_{\lambda}}\right\rangle_{q,t}$
		$\displaystyle=\left\langle p_{\pi}[Z],\sum_{\lambda\vdash m}q^{n(\lambda^{\prime})}t^{-2n(\lambda)}\frac{{\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}[X]{\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}[Z]{\mathfrak{J}}^{(\alpha,\gamma)}_{\lambda}[Y]}{\widetilde{j}^{(\alpha,\gamma)}_{\lambda}}\right\rangle_{q,t}$
		$\displaystyle=\sum_{\mu,\nu\vdash m}\bm{c}^{\pi}_{\mu,\nu}(\alpha,\gamma)p_{\mu}[X]p_{\nu}[Y].\qed$

5.7. Special cases in 7

In this subsection, we discuss some particular cases in 7, respectively related to marginals sums and to the specialization $q=t$ .

5.7.1. Marginal sums

We recall the usual $q$ -notation. We set

[m]_{q}:=1+q+\dots+q^{m-1}\text{ for any }m\geq 1,

[0]_{q}!=1\text{ and }[m]_{q}!:=[m]_{q}[m-1]_{q}\dots[1]_{q},\text{ for any }m\geq 1,

\text{and }\left[\begin{array}[]{c}m\\ k_{1},k_{2},\dots,k_{l}\end{array}\right]_{q}:=\frac{[m]_{q}!}{[k_{1}]_{q}!\dots[k_{l}]_{q}!}

for any $m,l\geq 0$ and $k_{1}+\dots+k_{l}=m$ . It is well known that all these quantities are polynomials in $q$ with non-negative integer coefficients. Finally, if $a$ is a parameter then

(a;q)_{m}:=(1-a)(1-qa)\cdots(1-q^{m-1}a)\text{ for any }m\geq 1.

Lemma 5.11.

Let $\lambda$ be a partition of size $m$ . Then,

\sum_{\mu\vdash m}\left[p_{\mu}\right]{J}^{(q,t)}_{\lambda}=J_{\lambda}[1]=\left\{\begin{array}[]{cc}(t;q)_{m}&\text{ if }\lambda=[m],\\ 0&otherwise.\end{array}\right.

Proof.

The first part of the equation is direct from the definitions. To obtain the second one we take $v=t$ in 2.2. ∎

For a proof of the following lemma, see [Mac95, Chapter VI, Section 8, Example 1].

Lemma 5.12.

For every $m\geq 1$ ,

{J}^{(q,t)}_{[m]}[X]=(q;q)_{m}\cdot h_{m}\left[X\frac{1-t}{1-q}\right]=(q;q)_{m}\sum_{\pi\vdash n}\frac{1}{z_{\pi}}p_{\pi}\left[X\frac{1-t}{1-q}\right].

Proposition 5.13.

Let $\pi,\mu\vdash m.$

\sum_{\nu\vdash m}\bm{c}^{\pi}_{\mu,\nu}(\alpha,\gamma)=(1-t)^{-m}\frac{1}{z_{\mu}}q^{\binom{m}{2}}(q;q)_{m}p_{\mu}\left[\frac{1-t}{1-q}\right].

Proof.

We adapt here the proof given in [GJ96a] for the Jack polynomials setting. Using 5.11, we write

\displaystyle\sum_{\nu\vdash m}\bm{c}^{\pi}_{\mu,\nu}(\alpha,\gamma)

\displaystyle=(1-t)^{-m}[p_{\mu}[Y]]\left\langle p_{\pi}[X],\sum_{\lambda\vdash m}t^{-2n(\lambda)}q^{n(\lambda^{\prime})}\frac{{J}^{(q,t)}_{\lambda}[X]{J}^{(q,t)}_{\lambda}[Y]{J}^{(q,t)}_{\lambda}[1]}{{j}^{(q,t)}_{\lambda}}\right\rangle_{q,t}.

But we know from 5.11 that only the term corresponding to $\lambda=[m]$ contributes to the sum. Hence,

	$\displaystyle\sum_{\nu\vdash m}\bm{c}^{\pi}_{\mu,\nu}(\alpha,\gamma)$	$\displaystyle=(1-t)^{-m}[p_{\mu}[Y]]\left\langle p_{\pi}[X],t^{-2n([m])}q^{n([m])}\frac{{J}^{(q,t)}_{[m]}[X]{J}^{(q,t)}_{[m]}[Y]{J}^{(q,t)}_{[m]}[1]}{{j}^{(q,t)}_{\lambda}}\right\rangle_{q,t}$
		$\displaystyle=(1-t)^{-m}[p_{\mu}[Y]]\left\langle p_{\pi}[X],q^{\binom{m}{2}}\frac{{J}^{(q,t)}_{[m]}[X]{J}^{(q,t)}_{[m]}[Y]\cdot(t;q)_{m}}{(q;q)_{m}(t;q)_{m}}\right\rangle_{q,t}$

We conclude using 5.12. ∎

Corollary 5.14.

For any $m\geq 1$ and $\pi,\mu,\nu\vdash m$ ,

z_{\mu}\sum_{\nu\vdash m}\bm{c}^{\pi}_{\mu,\nu}(\alpha,\gamma)

is a polynomial with non-negative integer coefficients in $\gamma$ and $\alpha$ .

Proof.

From 5.13, we get

	$\displaystyle z_{\mu}\sum_{\nu\vdash m}\bm{c}^{\pi}_{\mu,\nu}(\alpha,\gamma)$	$\displaystyle=\frac{q^{\binom{m}{2}}(q;q)_{m}}{(1-t)^{m}}\prod_{1\leq i\leq\ell(\mu)}\frac{1-t^{\mu_{i}}}{1-q^{\mu_{i}}}$
		$\displaystyle=\frac{q^{\binom{m}{2}}(q;q)_{m}}{(1-q)^{\ell(\mu)}(1-t)^{m-\ell(\mu)}}\prod_{1\leq i\leq\ell(\mu)}\frac{[\mu_{i}]_{t}}{[\mu_{i}]_{q}}$

But $(q;q)_{m}=[m]_{q}!(1-q)^{m}.$ Hence,

	$\displaystyle z_{\mu}\sum_{\nu\vdash m}\bm{c}^{\pi}_{\mu,\nu}(\alpha,\gamma)$	$\displaystyle=q^{\binom{m}{2}}[m]_{q}!\frac{(1-q)^{m-\ell(\mu)}}{(1-t)^{m-\ell(\mu)}}\prod_{1\leq i\leq\ell(\mu)}\frac{[\mu_{i}]_{t}}{[\mu_{i}]_{q}}$
		$\displaystyle=q^{\binom{m}{2}}[m]_{q}!\alpha^{m-\ell(\mu)}\prod_{1\leq i\leq\ell(\mu)}\frac{[\mu_{i}]_{t}}{[\mu_{i}]_{q}}.$

Note that $[m]!_{q}\prod_{1\leq i\leq\ell(m)}\frac{1}{[\mu_{i}]_{q}}$ is divisible by the binomial $\left[\begin{array}[]{c}m\\ \mu_{1},\mu_{2},\dots,\mu_{\ell(\mu)}\end{array}\right]_{q}.$ Hence, $[m]!_{q}\prod_{1\leq i\leq\ell(m)}\frac{1}{[\mu_{i}]_{q}}$ is a positive polynomial in $q$ , and by consequence in $\alpha$ and $\gamma$ . This finishes the proof. ∎

5.7.2. Integrality for the case $q=t$

When $q=t$ (equivalently $\alpha=1)$ , Macdonald polynomials are Schur functions up to a scalar factor, (see [Mac95, Chapter VI, Remark 8.4])

\mathfrak{J}^{(\alpha=1,\gamma)}_{\lambda}=H_{\lambda}(t)s_{\lambda},

where $t=\gamma+1$ as usual, and where

H_{\lambda}(t)=\prod_{\Box\in\lambda}[a_{\lambda}(\Box)+\ell_{\lambda}(\Box)+1]_{t}

is a $t$ -deformation of the hook product. Moreover, it follows from Eq. 6 and Eq. 43 that

\widetilde{j}^{(\alpha=1,\gamma)}_{\lambda}=H_{\lambda}(t)^{2}.

We recall that the expansion of Schur functions in the power-sum basis are given by the irreducible characters of the symmetric group $\chi^{\lambda}$ , see e.g [Mac95, Chapter I]

s_{\lambda}=\sum_{\mu\vdash|\lambda|}\frac{\chi^{\lambda}(\mu)}{z_{\mu}}p_{\mu}.

Hence we obtain the following formula for the coefficient of the generalized Matchings-Jack conjecture.

Proposition 5.15.

For any partitions $\pi,\mu$ and $\nu$ , one has

\bm{c}^{\pi}_{\mu,\nu}(\alpha=1,\gamma)=\frac{1}{z_{\mu}z_{\nu}}\sum_{\lambda\vdash|\pi|}t^{n(\lambda^{\prime})-2n(\lambda)}H_{\lambda}(t)\chi^{\lambda}(\pi)\chi^{\lambda}(\mu)\chi^{\lambda}(\nu).

We deduce the integrality of the coefficients in the parameter $\gamma$ .

Corollary 5.16.

For any partitions $\pi,\mu$ and $\nu$ of size $m$ , the normalized coefficient
$(1+\gamma)^{m(m-1)}z_{\mu}z_{\nu}\bm{c}^{\pi}_{\mu,\nu}(\alpha=1,\gamma)$ is a polynomial with integer coefficients in $\gamma$ .

Proof.

We use 5.15 and the fact that $n(\lambda^{\prime})-2n(\lambda)$ is minimal when $\lambda=[1^{m}]$ and the corresponding minimum is $m(m-1).$ ∎

References

[AF19] Per Alexandersson and Valentin Féray, A positivity conjecture on the structure constants of shifted Jack functions, preprint arXiv:1912.05203, (2019).
[BD22] Houcine Ben Dali, Generating series of non-oriented constellations and marginal sums in the Matching-Jack conjecture, Algebr. Comb. 5 (2022), no. 6, 1299–1336.
[BD23] by same author, Integrality in the Matching-Jack conjecture and the Farahat-Higman algebra, Transactions of the American Mathematical Society 376 (2023), no. 05, 3641–3662.
[BDD23] Houcine Ben Dali and Maciej Dołęga, Positive formula for Jack polynomials, Jack characters and proof of Lassalle’s conjecture, preprint arXiv:2305.07966, (2023).
[BGHT99] François Bergeron, Adriano M Garsia, Mark Haiman, and Glenn Tesler, Identities and positivity conjectures for some remarkable operators in the theory of symmetric functions, Methods and applications of analysis 6 (1999), no. 3, 363–420.
[BHIR23] François Bergeron, Jim Haglund, Alessandro Iraci, and Marino Romero, The super nabla operator, preprint arXiv:2303.00560, (2023).
[CD22] Guillaume Chapuy and Maciej Dołęga, Non-orientable branched coverings, $b$ -Hurwitz numbers, and positivity for multiparametric Jack expansions, Adv. Math. 409 (2022), no. part A, Paper No. 108645, 72. MR 4477016
[CDM23] Cesar Cuenca, Maciej Dołęga, and Alexander Moll, Universality of global asymptotics of Jack-deformed random Young diagrams at varying temperatures, Preprint arXiv:2304.04089, 2023.
[CM18] Erik Carlsson and Anton Mellit, A proof of the shuffle conjecture, J. Amer. Math. Soc. 31 (2018), no. 3, 661–697. MR 3787405
[DF16] Maciej Dołęga and Valentin Féray, Gaussian fluctuations of Young diagrams and structure constants of Jack characters, Duke Math. J. 165 (2016), no. 7, 1193–1282. MR 3498866
[DF17] Maciej Dołęga and Valentin Féray, Cumulants of Jack symmetric functions and the $b$ -conjecture, Trans. Amer. Math. Soc. 369 (2017), no. 12, 9015–9039. MR 3710651
[DIVW21] Michele D’Adderio, Alessandro Iraci, and Anna Vanden Wyngaerd, Theta operators, refined delta conjectures, and coinvariants, Adv. Math. 376 (2021), Paper No. 107447, 59. MR 4178919
[DM22] Michele D’Adderio and Anton Mellit, A proof of the compositional Delta conjecture, Advances in Mathematics 402 (2022), 108342.
[DR23] Michele D’Adderio and Marino Romero, New identities for Theta operators, Transactions of the American Mathematical Society 376 (2023), no. 8, 5775–5807.
[GH93] A. Garsia and M. Haiman, A graded representation model for Macdonald’s polynomials, Proc. Nat. Acad. Sci. U.S.A. 90 (1993), no. 8, 3607–3610. MR 1214091
[GHT01] Adriano M Garsia, Mark Haiman, and Glenn Tesler, Explicit plethystic formulas for Macdonald q, t-Kostka coefficients, The Andrews Festschrift: Seventeen Papers on Classical Number Theory and Combinatorics, Springer, 2001, pp. 253–297.
[GJ96a] I. P. Goulden and D. M. Jackson, Connection coefficients, matchings, maps and combinatorial conjectures for Jack symmetric functions, Trans. Amer. Math. Soc. 348 (1996), no. 3, 873–892. MR 1325917 (96m:05196)
[GJ96b] by same author, Maps in locally orientable surfaces, the double coset algebra, and zonal polynomials, Canad. J. Math. 48 (1996), no. 3, 569–584. MR 1402328 (97h:05051)
[GM19] Adriano Garsia and Anton Mellit, Five-term relation and Macdonald polynomials, Journal of Combinatorial Theory, Series A 163 (2019), 182–194.
[GT96] A. M. Garsia and G. Tesler, Plethystic formulas for Macdonald $q,t$ -Kostka coefficients, Adv. Math. 123 (1996), no. 2, 144–222. MR 1420484
[Hai02] Mark Haiman, Vanishing theorems and character formulas for the hilbert scheme of points in the plane, Inventiones mathematicae 149 (2002), 371–407.
[HHL05] J. Haglund, M. Haiman, and N. Loehr, A combinatorial formula for Macdonald polynomials, J. Amer. Math. Soc. 18 (2005), no. 3, 735–761. MR 2138143
[HRW18] J. Haglund, J. B. Remmel, and A. T. Wilson, The delta conjecture, Trans. Amer. Math. Soc. 370 (2018), no. 6, 4029–4057. MR 3811519
[Jac71] Henry Jack, A class of symmetric polynomials with a parameter, Proc. Roy. Soc. Edinburgh Sect. A 69 (1970/1971), 1–18. MR MR0289462 (44 #6652)
[Kno97] Friedrich Knop, Symmetric and non-symmetric quantum Capelli polynomials, Comment. Math. Helv. 72 (1997), no. 1, 84–100.
[KO94] Sergei Kerov and Grigori Olshanski, Polynomial functions on the set of Young diagrams, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), no. 2, 121–126. MR MR1288389 (95f:05116)
[KS97] Friedrich Knop and Siddhartha Sahi, A recursion and a combinatorial formula for Jack polynomials, Invent. Math. 128 (1997), no. 1, 9–22. MR 1437493
[Las98] Michel Lassalle, Coefficients binomiaux généralisés et polynômes de Macdonald, J. Funct. Anal. 158 (1998), no. 2, 289–324.
[Las08] by same author, A positivity conjecture for Jack polynomials, Math. Res. Lett. 15 (2008), no. 4, 661–681. MR 2424904
[Mac95] I. G. Macdonald, Symmetric functions and Hall polynomials, second ed., Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995, With contributions by A. Zelevinsky, Oxford Science Publications. MR 1354144
[Oko98] Andrei Okounkov, (Shifted) Macdonald polynomials: $q$ -integral representation and combinatorial formula, Compositio Math. 112 (1998), no. 2, 147–182. MR MR1626029 (99h:05120)
[Sah96] Siddhartha Sahi, Interpolation, integrality, and a generalization of Macdonald’s polynomials, Internat. Math. Res. Notices 1996 (1996), no. 10, 457–471.
[Śni19] Piotr Śniady, Asymptotics of Jack characters, J. Combin. Theory Ser. A 166 (2019), 91–143. MR 3921039
[Sta89] Richard P. Stanley, Some combinatorial properties of Jack symmetric functions, Adv. Math. 77 (1989), no. 1, 76–115. MR 1014073 (90g:05020)

Macdonald characters from a new formula for Macdonald polynomials

Abstract.

1. Introduction

1.1. Jack and Macdonald polynomials

1.2. Symmetric functions and plethysm

1.3. A new formula for Macdonald polynomials

Theorem 1.1.

Theorem 1.2.

1.4. Shifted symmetric functions and Macdonald characters

Definition 1.3.

Definition 1.4.

Theorem 1.5.

1.5. A new parametrization and Macdonald version of some Jack conjectures

1.6. Outline of the paper

1.7. Acknowledgments

2. Preliminaries

2.1. Partitions

2.2. Integral form of Macdonald polynomials:

Theorem 2.1.

Theorem 2.2.

Theorem 2.3 (Pieri rule).

3. A new creation formula for Macdonald polynomials

3.1. Modified Macdonald polynomials

Remark 1.

3.2. Creation formula for modified Macdonald polynomials

Remark 2.

Lemma 3.1.

Proof.

Theorem 3.2.

Proof.

3.3. Vanishing property and second creation formula

Theorem 3.3.

Theorem 3.4 (Pieri rule for H~λ(q,t)\widetilde{H}^{(q,t)}_{\lambda}).

Remark 3.

Proposition 3.5.

Proof.

Proof of 3.3.

Remark 4.

3.4. Proof of Theorems 1.2 and 1.1

Lemma 3.6.

Proof.

Lemma 3.7.

Proof.

4. Macdonald characters

4.1. Shifted symmetric Macdonald polynomials

Theorem 4.1 (Shifted Macdonald polynomials).

Remark 5.

4.2. An explicit isomorphism between the spaces of symmetric and shifted-symmetric functions

Lemma 4.2.

Proof.

Theorem 4.3.

Proof.

Remark 6.

4.3. Macdonald characters are shifted symmetric

Proof of 1.5.

Corollary 4.4.

Proof.

4.4. Characterization theorem

Theorem 4.5.

Lemma 4.6.

Proof.

Proof of 4.5.

4.5. Positivity conjectures about the characters 𝜽~μ(q,t)\bm{\widetilde{\theta}}^{(q,t)}_{\mu}

Conjecture 1.

Proposition 4.7.

Proof.

5. Macdonald versions for some Jack conjectures

5.1. Jack polynomials and a new normalization of Macdonald polynomials

Proposition 5.1.

Proof.

5.2. Stanley’s conjecture

5.2.1. Jack case

Conjecture 2.

5.2.2. Macdonald generalization

Conjecture 3 (Macdonald version of Stanley’s conjecture).

5.3. Lassalle’s conjecture

5.3.1. Jack case

Definition 5.2 (Jack characters).

Theorem 5.3 ([BDD23]).

5.3.2. Macdonald generalization

Theorem 3.4 (Pieri rule for $\widetilde{H}^{(q,t)}_{\lambda}$ ).

4.5. Positivity conjectures about the characters $\bm{\widetilde{\theta}}^{(q,t)}_{\mu}$

Conjecture 6 ( $b$ -conjecture [GJ96a]).

Conjecture 8 (A Macdonald generalization of the $b$ -conjecture).

5.5. Connection with the Structure coefficients $\bm{g}^{\pi}_{\mu,\nu}$ of Macdonald characters

5.7.2. Integrality for the case $q=t$