2020-09-09

## Macdonald P polynomials

Macdonald polynomials were introduced by I.G. Macdonald in [Mac88]. It is a two-parameter extension of the Schur functions, and unify the Jack polynomials and Hall-Littlewood polynomials.

R. Langer's master thesis [Lan09] from 2009 gives a nice overview.

### Inner product characterization

Let $\langle \cdot , \cdot \rangle_{q,t}$ denote the inner product on symmetric functions such that

\[ \langle \powerSum_\lambda , \powerSum_\mu \rangle_{q,t} = z_\lambda \delta_{\lambda\mu} \prod_{i=1}^{\length{\lambda}} \frac{1-q^{\lambda_i}}{1-t^{\lambda_i}} . \]Note that at $q=t=0,$ we recover the standard Hall inner product.

The Macdonald polynomials $\macdonaldP_\lambda(\xvec;q,t)$ are defined as the unique family of polynomials such that

\begin{equation*} \macdonaldP_\lambda(\xvec;q,t) = \monomial_\lambda(\xvec) + \sum_{\mu \prec \lambda} \eta_{\lambda\mu} \monomial_\mu(\xvec) \end{equation*}and $\langle \macdonaldP_\lambda, \macdonaldP_\mu \rangle_{q,r}=0$ whenever $\lambda \neq \mu.$ Here, $\prec$ denotes the dominance order on partitions.

**Example (Macdonald polynomials for $\lambda \vdash 3$).**

We have the following Macdonald polynomials:

\begin{align*} \macdonaldP_{111}(\xvec;q,t) &= \monomial_{111} \\ \macdonaldP_{21}(\xvec;q,t) &= \monomial_{21} + \frac{(t-1) (2 q t+q+t+2)}{q t^2-1} \monomial_{111} \\ \macdonaldP_{3}(\xvec;q,t) &= \monomial_{3} + \frac{\left(q^2+q+1\right)(t-1)}{q^2 t-1} \monomial_{21} \\ &\phantom{=}+ \frac{(q+1) \left(q^2+q+1\right) (t-1)^2}{(q t-1) \left(q^2 t-1\right)} \monomial_{111} \end{align*}Note that this characterization is quite useless for actually computing Macdonald polynomials, it is really inefficient. The tableau formula below is much faster.

### Eigenvector characterization

The Macdonald polynomials can also be characterized as eigenvectors to a certain operator $D,$ see [Mac88]. The operator $D$ is given as

\[ D = \sum_{i=1}^n \prod_{j \neq i} \left( \frac{tx_i-x_j}{x_i-x_j} \right) T_{q,x_i} \]where $T_{q,x_i} f(x_1,\dotsc,x_n) = f(x_1,\dotsc,x_{i-1},qx_i,x_{i+1},\dotsc,x_n).$

There is also a way to produce Macdonald polynomials via determinants, see [LLM98].

### Tableau formula

There is a quite cumbersome way to express Macdonald polynomials as a sum over tableaux, see [Mac88] and [Mac95]. We first need to introduce some notation. Given $\lambda/\mu,$ let $R_{\lambda/\mu}$ and $C_{\lambda/\mu}$ be the set of rows (columns, resp.) containing some box of $\lambda/\mu.$

For $\lambda/\mu$ being a *horizontal strip* (no two boxes in the same column),
we define $\psi_{\lambda/\mu}(q,t)$ as the following product.
Set

where

\[ b_{\mu}(s;q,t) \coloneqq \begin{cases} \frac{ 1-q^{\arm_\mu(s)}t^{1+\leg_\mu(s)} }{ 1-q^{1+\arm_\mu(s)}t^{\leg_\mu(s)} } &\text{if $s\in \mu$} \\ 1 &\text{otherwise}. \end{cases} \]**Example (Computation of $\psi_{\lambda/\mu}(q,t)$).**

Let $\lambda=(8,5,5,1)$ and $\mu=(6,5,2).$ One can fairly easy see that it suffices to take the product over only the set of boxes both in $\mu$ and in $R_{\lambda/\mu} \setminus C_{\lambda/\mu}.$ Here there are three such boxes, $(1,2),$ $(1,6)$ and $(3,2),$ marked in the figure.

$\ast$ | $\ast$ | ||||||

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We then have

\[ b_{\mu}((1,2);q,t) = \frac{1-q^{4}t^{1+2}}{1-q^{1+4}t^{2}} \qquad b_{\lambda}((1,2);q,t) = \frac{1-q^{6}t^{1+2}}{1-q^{1+6}t^{2}}. \] \[ b_{\mu}((1,6);q,t) = \frac{1-q^{0}t^{1+0}}{1-q^{1+0}t^{0}} \qquad b_{\lambda}((1,6);q,t) = \frac{1-q^{2}t^{1+0}}{1-q^{1+2}t^{0}}. \] \[ b_{\mu}((3,2);q,t) = \frac{1-q^{0}t^{1+0}}{1-q^{1+0}t^{0}} \qquad b_{\lambda}((3,2);q,t) = \frac{1-q^{3}t^{1+0}}{1-q^{1+3}t^{0}}. \]In total, $\psi_{\lambda/\mu}(q,t)$ is given by

\[ \frac{(1-q^{4}t^{3})}{(1-q^{5}t^{2})} \frac{(1-q^{7}t^{2})}{(1-q^{6}t^{3})} \frac{(1-t)}{(1-q)} \frac{(1-q^{3})}{(1-q^{2}t)} \frac{(1-t)}{(1-q)} \frac{(1-q^{4})}{(1-q^{3}t)}. \]Now, if $T$ is a semi-standard Young tableau, we let

\[ \psi_{T}(q,t) \coloneqq \prod_{j=1}^n \psi_{\lambda^{j}/\lambda^{j-1}}(q,t) \]where $\lambda^{j}/\lambda^{j-1}$ is the horizontal strip determined by the entries with value $j$ in $T.$

Finally, we have that for $\mu \vdash n$

\[ \macdonaldP_{\mu}(\xvec;q,t) = \sum_{\nu \vdash n} \monomial_\nu(\xvec) \sum_{T \in \SSYT(\mu,\nu)} \psi_{T}(q,t). \]Note that if $q=t,$ this formula does indeed give the Schur polynomial $\schur_\mu(\xvec).$

### Properties

The Macdonald polynomials specialize to other families of symmetric functions, the Schur, Hall-Littlewood and Jack polynomials:

\begin{equation*} \schurS_\lambda(\xvec) = \macdonaldP_\lambda(\xvec;q,q), \qquad \hallLittlewoodP_\lambda(\xvec;t) = \macdonaldP_\lambda(\xvec;0,t) \qquad \jackP_\lambda(\xvec;a) = \lim_{t \to 1} \macdonaldP_\lambda(\xvec;t^a,t) \end{equation*}### qt-Koskta coefficients

Conider the expansion

\[ \schurS_\lambda = \sum_{\mu} K_{\lambda\mu}(q,t) \macdonaldP_\mu(\xvec;q,t). \]In [Mac88], it is conjectured that the $K_{\lambda\mu}(q,t)$ are polynomials in $\setP[q,t].$ This was later confirmed by M. Haiman,[Hai01], as this property follows from the Schur positivity of the modified Macdonald polynomials.

## Macdonald J polynomials

By rescaling, the integral form Macdonald polynomials are defined via

\begin{equation*} \macdonaldJ_\lambda(x;q,t) = \left( \prod_{\square \in \lambda} 1-q^{\arm(\square)}t^{1+\leg(\square)} \right) \macdonaldP_\lambda(x;q,t). \end{equation*}Similarly, the integral-form Jack polynomials are obtained as a limit:

\[ \jackJ_\lambda(x;a) = \lim_{t \to 1} \frac{\macdonaldJ_\lambda(x;t^a,t)}{(1-t)^{|\lambda|}} \]The modified Macdonald polynomials (also known as transformed Macdonald polynomials) are obtained by letting

\begin{equation*} \macdonaldH_\lambda(x;q,t) := t^{n(\lambda)}H_\lambda(x;q,1/t),\qquad \text{ where } \qquad H_\lambda(x;q,t) = \macdonaldJ_\lambda[x/(1-t);q,t], \end{equation*}and we use plethystic notation in the last expression.

## Multi-Macdonald P polynomials

In [GL19b], C. González and L. Lapointe introduce the multi-Macdonald polynomials, which are indexed by $r$-tuples of partitions. They are defined via triangularity relations, similar to the definition of Macdonald P polynomials. When $r=1,$ the usual Macdonald P polynomials are recovered, and when $r=2,$ the family coincide with the double Macdonald polynomials, previously defined in [BLM14].

The multi-Macdonald polynomials can be expressed as a product of Macdonald-P polynomials, with suitable plethystic substitutions in each factor.

There is an analog of the modified Macdonald polynomials, and then an analog of the $qt$-Kostka polynomials when expanded in the Wreath product Schur polynomials. The authors show that the multi-$qt$-Kostka polynomials are in $\setN[q,t].$

## References

- [BLM14] O. Blondeau-Fournier, L. Lapointe and P. Mathieu. Double Macdonald polynomials as the stable limit of Macdonald superpolynomials. Journal of Algebraic Combinatorics, 41(2):397–459, July 2014.
- [GL19b] Camilo González and Luc Lapointe. Multi-Macdonald polynomials. arXiv e-prints, 2019.
- [Hai01] Mark D. Haiman. Hilbert schemes, polygraphs and the Macdonald positivity conjecture. Journal of the American Mathematical Society, 14(04):941–1007, October 2001.
- [Lan09] Robin Langer. Macdonald polynomials and symmetric functions. arXiv e-prints, 2009.
- [LLM98] Luc Lapointe, Alain Lascoux and Jennifer Morse. Determinantal expressions for Macdonald polynomials. International Mathematics Research Notices, 1998(18):957, 1998.
- [Mac88] Ian G. Macdonald. A new class of symmetric functions. Séminaire Lotharingien de Combinatoire [electronic only], 20(B20a), 1988.
- [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