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2020-04-09

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.

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 \[ \schurS_{4}(\xvec) + (q+t+t^2) \schurS_{31}(\xvec)+(q t+t^2) \schurS_{22}(\xvec) +(q t+q t^2+t^3) \schurS_{211}(\xvec)+ q t^3 \schurS_{1111}(\xvec) \]

Combinatorial formula II

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 $\mu.$

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 coefficients $\tilde{K}_{\lambda\mu}(q,t)$ — also called $qt$-Kostka coefficients — are elements in $\setN[q,t].$ These coefficients 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 coefficients are $\tilde{K}_{\lambda\mu}(q,t) = t^{n(\mu)}K_{\lambda \mu}(q,1/t)$

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}

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.

There is also an analog in defined in https://www.newton.ac.uk/files/preprints/ni01035.pdf.

References