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:

$\;$ | $\textbf{4}$ | $\textbf{31}$ | $\textbf{22}$ | $\textbf{211}$ | $\textbf{1111}$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

$\textbf{4}$ | $1$ | $t^3+t^2+t$ | $t^4+t^2$ | $t^5+t^4+t^3$ | $t^6$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

$\textbf{31}$ | $1$ | $q+t^2+t$ | $q t+t^2$ | $q t^2+q t+t^3$ | $q t^3$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

$\textbf{22}$ | $1$ | $q t+q+t$ | $q^2+t^2$ | $q^2 t+q t^2+q t$ | $q^2 t^2$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

$\textbf{211}$ | $1$ | $q^2+q+t$ | $q^2+q t$ | $q^3+q^2 t+q t$ | $q^3 t$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

$\textbf{1111}$ | $1$ | $q^3+q^2+q$ | $q^4+q^2$ | $q^5+q^4+q^3$ | $q^6$ | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||

### 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.

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

## References

- [Ban07] Jason Bandlow. Combinatorics of Macdonald polynomials and extensions. UC San Diego. 2007.
- [BZ05] Nantel Bergeron and Michael Zabrocki. q and $q,t$-analogs of non-commutative symmetric functions. Discrete Mathematics, 298(1-3):79–103, August 2005.
- [CHMMW22] Sylvie Corteel, Jim Haglund, Olya Mandelshtam, Sarah Mason and Lauren Williams. Compact formulas for Macdonald polynomials and quasisymmetric Macdonald polynomials. Selecta Mathematica, 28(2), January 2022.
- [Doł19] Maciej Dołęga. Macdonald cumulants, G-inversion polynomials and G-parking functions. European Journal of Combinatorics, 75:172–194, January 2019.
- [Gil16] Maria Monks Gillespie. A combinatorial approach to the $q,t$-symmetry relation in Macdonald polynomials. The Electronic Journal of Combinatorics, 23(2), May 2016.
- [GT96] A.M. Garsia and G. Tesler. Plethystic formulas for Macdonald $qt$-Kostka coefficients. Advances in Mathematics, 123(2):144–222, November 1996.
- [Hag07] James Haglund. The $q,t$-Catalan numbers and the space of diagonal harmonics (University lecture series). American Mathematical Society, 2007.
- [Hai01] Mark D. Haiman. Hilbert schemes, polygraphs and the Macdonald positivity conjecture. Journal of the American Mathematical Society, 14(04):941–1007, October 2001.
- [HHL05] James Haglund, Mark D. Haiman and Nicholas A. Loehr. A combinatorial formula for Macdonald polynomials. J. Amer. Math. Soc., 18(03):735–762, July 2005.
- [KM19] Ryan Kaliszewski and Jennifer Morse. Colorful combinatorics and Macdonald polynomials. European Journal of Combinatorics, 81:354–377, October 2019.
- [LOR22] Seung Jin Lee, Jaeseong Oh and Brendon Rhoades. Two conjectures for Macdonald polynomials: the stretching symmetry and haglund's conjecture. arXiv e-prints, 2022.
- [Mac95] Ian G. Macdonald. Symmetric functions and Hall polynomials. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, Second edition, 1995. With contributions by A. Zelevinsky, Oxford Science Publications