# The symmetric functions catalog

An overview of symmetric functions and related topics

2022-06-06

## Modified Macdonald polynomials

The modified (or transformed) Macdonald polynomials $\macdonaldH_{\lambda}(\xvec;q,t)$ appear as a combinatorial version of the Macdonald $P$ polynomials. The family is indexed by partitions $\lambda$ and they are symmetric in $\xvec.$ It was proved by M. Haiman in [Hai01] that the modified Macdonald polynomials are bigraded Frobenius characteristics of certain $\symS_n$-modules that appear in diagonal harmonics. For details on this story, see J. Swanson's notes from the n! conjecture seminar.

A combinatorial formula for the modified Macdonald polynomials was proved in [HHL05], where a close connection with LLT polynomials is made apparent. The canonical reference on modified Macdonald polynomials is the book by Jim Haglund, [Hag07]. It is a major open problem in algebraic combinatorics to give a combinatorial proof (by using RSK, dual equivalence or crystals) that the $\macdonaldH_{\lambda}(\xvec;q,t)$ are Schur-positive.

### Definition

The modified Macdonald polynomials were originally defined via the Macdonald J polynomials, as

$\macdonaldH_{\mu}(\xvec;q,t) = t^{n(\mu)} \macdonaldJ[ X/(1-t^{-1}) ;q,1/t].$

Note that we use plethystic notation here.

### Inner product characterization

The modified Macdonald polynomials $\{\macdonaldH_{\mu} \}_{\mu \vdash n}$ is the unique family of symmetric functions with coefficients in $\setQ(q,t),$ such that

• $\macdonaldH_{\mu}[X(1-q);q,t] \in \setQ(q,t)\{ \schurS_\lambda : \lambda \trianglerighteq \mu \}$
• $\macdonaldH_{\mu}[X(1-t);q,t] \in \setQ(q,t)\{ \schurS_\lambda : \lambda \trianglerighteq \mu' \}$
• $\langle \macdonaldH_{\mu} , \schurS_\mu \rangle = 1.$

Alternatively, they are the the unique family of polynomials that fulfills

• $\langle \macdonaldH_{\mu}[X;q,t], \schurS_\lambda[X/(t-1)] \rangle =0$ whenever $\lambda \triangleright \mu ,$
• $\langle \macdonaldH_{\mu}[X;q,t], \schurS_\lambda[X/(1-q)] \rangle =0$ whenever $\lambda \triangleleft \mu ,$
• $\langle \macdonaldH_{\mu} , \schurS_{(n)} \rangle = 1.$

### Haglund's combinatorial formula

A combinatorial formula for the modified Macdonald polynomials was obtained in [HHL05]. It is given by

$\macdonaldH_{\lambda}(\xvec;q,t) = \sum_{T:\lambda \to \setN} q^{\inv_\lambda(T)} t^{\maj_\lambda(T)} \xvec^T,$

where $\inv_\lambda(T)$ and $\maj_\lambda(T)$ are certain combinatorial statistics.

I highly recommend reading The genesis of the Macdonald polynomial statistic by Jim Haglund.

Example (Macdonald polynomials for $|\lambda|=4.$).

The modified Macdonald polynomial $\macdonaldH_{\lambda}(\xvec;q,t)$ have the following Schur expansions:

 $\;$ $\textbf{4}$ $\textbf{31}$ $\textbf{22}$ $\textbf{211}$ $\textbf{1111}$ $\textbf{4}$ $1$ $t^3+t^2+t$ $t^4+t^2$ $t^5+t^4+t^3$ $t^6$ $\textbf{31}$ $1$ $q+t^2+t$ $q t+t^2$ $q t^2+q t+t^3$ $q t^3$ $\textbf{22}$ $1$ $q t+q+t$ $q^2+t^2$ $q^2 t+q t^2+q t$ $q^2 t^2$ $\textbf{211}$ $1$ $q^2+q+t$ $q^2+q t$ $q^3+q^2 t+q t$ $q^3 t$ $\textbf{1111}$ $1$ $q^3+q^2+q$ $q^4+q^2$ $q^5+q^4+q^3$ $q^6$
For example, $\macdonaldH_{31}(\xvec;q,t)$ is given by $\schurS_{4}(\xvec) + (q+t+t^2) \schurS_{31}(\xvec)+(q t+t^2) \schurS_{22}(\xvec) +(q t+q t^2+t^3) \schurS_{211}(\xvec)+ q t^3 \schurS_{1111}(\xvec)$

### Alternative combinatorial formulas

A second combinatorial formula is given by R. Kaliszewski and J. Morse in [KM19]. We have

$\macdonaldH_{\lambda}(\xvec;q,t) = \sum_{T} q^{betrayal(T)} t^{\cocharge(T)} \xvec^T,$

where the sum is over all tabloids with content $\mu.$

A more recent formula is given in [CHMMW22], where certain terms are grouped together:

$\macdonaldH_{\lambda'}(\xvec;q,t) = \sum_{\sigma \in \mathrm{ST}(\lambda)} \xvec^\sigma t^{\inv_\lambda(\sigma)} q^{\maj_\lambda(\sigma)} \mathrm{perm}_t(\sigma,\lambda)$

where $\mathrm{perm}_t(\sigma,\lambda)$ is a certain $t$-multinomial coefficient, and $\mathrm{ST}(\lambda)$ is the set of sorted tableaux of shape $\lambda.$ In some sense, certain monomial terms in the original formula have been grouped together in order to produce multinomial coefficients.

### Properties

We have the following specializations:

$\macdonaldH_{\lambda}(\xvec;0,0) = \completeH_{n}(\xvec), \qquad \macdonaldH_{\lambda}(\xvec;1,0) = \prod_i \completeH_{\lambda_i}(\xvec), \qquad \macdonaldH_{\lambda}(\xvec;1,1) = (\elementaryE_1(\xvec))^{|\lambda|}, \qquad$

We have the following symmetries:

$\macdonaldH_{\lambda}(\xvec;q,t) = \macdonaldH_{\lambda'}(\xvec;t,q) = q^{n(\lambda)} t^{n(\lambda')}\omega \macdonaldH_{\lambda}(\xvec;q^{-1},t^{-1})$

where $n(\lambda) = \sum_i (n-i)\lambda(i)$ when $\lambda$ has size $n.$ The first identity follows immediately from the fact that the modified Macdonald polynomials is a bigraded Frobenius series with this symmetry. However, a bijective proof is not known. Some partial results have been found by M. Gillespie, [Gil16].

The modified Macdonald polynomials specialize to the modified Hall–Littlewood polynomials at $q=0,$ and these are in turn closely related to the transformed Hall–Littlewood polynomials.

### Schur expansion

M. Haiman defines a family of bigraded $\symS_n$-modules, which has the property that the Frobenius image of this family gives the modified Macdonald polynomials, see [Hai01]. It follows that the modified Macdonald polynomials are Schur positive, although no combinatorial formula is known.

Problem (Macdonald, [Mac95]).

Find pairs of statistics $\maj_\mu(\cdot)$ and $\inv_\mu(\cdot)$ on standard Young tableaux, such that

$\macdonaldH_{\mu}(\xvec;q,t) = \sum_{T \in \SYT(n)} q^{\inv_\mu(T)} t^{\maj_\mu(T)} \schurS_{sh(T)}(\xvec).$

This would give a combinatorial proof that the modified Macdonald–Kostka polynomials $\tilde{K}_{\lambda\mu}(q,t)$ — also called $qt$-Kostka polynomials — are elements in $\setN[q,t].$ These polynomials are defined via the relation

$\macdonaldH_{\mu}(\xvec;q,t) = \sum_{\lambda} \tilde{K}_{\lambda\mu}(q,t) \schurS_{\lambda}(\xvec).$

Note that $\tilde{K}_{\lambda\mu}(1,1) = f^\lambda,$ the number of standard Young tableaux.

### $qt$-Kostka polynomials

The following properties can be found in [p.32, Hag07].

We have that the modified Kostka polynomials are $\tilde{K}_{\lambda\mu}(q,t) = t^{n(\mu)}K_{\lambda \mu}(q,1/t),$ where the $K_{\lambda \mu}$ are the $qt$-Kostka polynomials.

Recall that the $qt$-Kostka polynomials $K_{\lambda \mu}(q,t)$ have the following properties:

$K_{\lambda \mu}(0,t) = K_{\lambda \mu}(t) = \sum_{T \in \SSYT(\lambda,\mu)} t^{\charge(T)}.$

and thus $K_{\lambda \mu}(0,1) = K_{\lambda \mu}.$ We also define the cocharge polynomial,

$\tilde{K}_{\lambda \mu}(t) = t^{n(\mu)}K_{\lambda \mu}(1/t) \sum_{T \in \SSYT(\lambda,\mu)} t^{\cocharge(T)}.$

Also, $\tilde{K}_{\lambda\mu}(1,1) = K_{\lambda\mu}(1,1) = f^{\lambda},$ the number of standard Young tableaux of shape $\lambda.$

Then there are the following symmetries:

\begin{align} K_{\lambda,\mu}(q,t) &= t^{n(\mu)} q^{n(\mu')} K_{\lambda',\mu}(1/q,1/t) \\ K_{\lambda,\mu}(q,t) &= K_{\lambda',\mu'}(t,q)\\ % \tilde{K}_{\lambda',\mu}(q,t) &= t^{n(\mu)} q^{n(\mu')} \tilde{K}_{\lambda',\mu}(1/q,1/t) \\ \tilde{K}_{\lambda,\mu}(q,t) &= \tilde{K}_{\lambda,\mu'}(t,q)\\ \end{align}

### Stretching symmetry property

The following property is stated as a conjecture in [LOR22], but it was later revealed that this is a known property, following from a result by Garsia–Tesler[Thm I.1, GT96].

Proposition.

Let $k$ be a positive integer. Then

$\macdonaldH_{k \mu}(\xvec;q,q^k) = \macdonaldH_{k \mu'}(\xvec;q,q^k).$

That is, we have a type of symmetry under conjugation and stretching.

### General diagrams

In [Ban07], J. Bandlow studies more general shapes, in particular skew shapes.

### Macdonald cumulants

A generalization of the $qt$-Kostka coefficients are defined in [Doł19], which are related to a certain Schur-positivity problem, generalizing the Schur positivity of the modified Macdonald polynomials.

### Non-commutative Macdonald polynomials

In [BZ05], the authors introduce a non-commutative family of symmetric functions with properties similar to those of the modified Macdonald polynomials.