The symmetric functions catalog

An overview of symmetric functions and related topics

2022-07-26

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

\[ \psi_{\lambda/\mu}(q,t) \coloneqq \prod_{\substack{ s \in R_{\lambda/\mu} \setminus C_{\lambda/\mu} }} \frac{b_{\mu}(s;q,t)}{b_{\lambda}(s;q,t)} \]

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.

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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 $\schurS_\mu(\xvec).$

From non-symmetric Macdonald E polynomials

In [Prop. 5.3.1, HHL08], the following expansion is obtained:

\begin{equation*} \macdonaldP_\lambda(\xvec;q,t) = \prod_{u \in \lambda} \left(1- q^{1+\leg(u)}t^{\arm(u)}\right) \sum_{\gamma \sim \lambda} \frac{ \macdonaldE_{\gamma}(x_1,\dotsc,x_n;q^{-1},t^{-1})}{ \prod_{v \in \gamma} \left(1- q^{1+\leg(v)}t^{\arm(v)}\right) }. \end{equation*}

The polynomial $\macdonaldE_{\gamma}(\xvec;q,t)$ is a non-symmetric Macdonald polynomial.

We can express this formula using permuted-basement Macdonald polynomials also,

\begin{equation*} \macdonaldP_\lambda(\xvec;q,t) = \prod_{u \in \lambda} \left(1- q^{1+\leg(u)}t^{\arm(u)}\right) \sum_{\gamma \sim \lambda} \frac{ \macdonaldE^{id}_{\gamma}(x_1,\dotsc,x_n;q,t)}{ \prod_{v \in \gamma} \left(1- q^{1+\leg(v)}t^{\arm(v)}\right) }, \end{equation*}

since we have the relation $\macdonaldE^{id}_{\gamma}(x_1,\dotsc,x_n;q,t) = \macdonaldE_{\gamma}(x_n,\dotsc,x_1;q^{-1},t^{-1}).$

In fact, an analog of this formula exists for any fixed basement, see [Thm. 29, Ale19a].

Specializations

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*}

Macdonald J polynomials

By rescaling, the integral form Macdonald polynomials are defined via

\begin{equation*} \macdonaldJ_\lambda(\xvec;q,t) = \left( \prod_{\square \in \lambda} 1-q^{\arm(\square)}t^{1+\leg(\square)} \right) \macdonaldP_\lambda(\xvec;q,t). \end{equation*}

Similarly, the integral-form Jack polynomials are obtained as a limit:

\[ \jackJ_\lambda(\xvec;a) = \lim_{t \to 1} \frac{\macdonaldJ_\lambda(\xvec;t^a,t)}{(1-t)^{|\lambda|}} \]

The modified Macdonald polynomials (also known as transformed Macdonald polynomials) are obtained by letting

\begin{equation*} \macdonaldH_\lambda(\xvec;q,t) := t^{n(\lambda)}H_\lambda(\xvec;q,1/t),\qquad \text{ where } \qquad H_\lambda(\xvec;q,t) = \macdonaldJ_\lambda[\xvec/(1-t);q,t], \end{equation*}

and we use plethystic notation in the last expression.

qt-Koskta polynomials

Macdonald [p.354, Mac88] introduces the $qt$-Kostka polynomials $K_{\lambda\mu}(q,t)$ via the relation

\[ \macdonaldJ_\mu(\xvec;q,t) = \sum_{\lambda} K_{\lambda\mu}(q,t) \schurS_\lambda[\xvec(1-t)] \]

where

\[ \schurS_\lambda[\xvec(1-t)] = \sum_{\rho} \frac{\chi_\rho^\lambda}{z_\rho} \powerSum_\rho(x) \prod_{i=1}^{\length(\rho)} (1-t^{\rho_i}). \]

In [Mac88], it is conjectured that the $K_{\lambda\mu}(q,t)$ are polynomials in $\setN[q,t].$ This was later confirmed by M. Haiman [Hai01], as this property follows from the Schur positivity of the modified Macdonald polynomials, and the corresponding modified $qt$-Kostka polynomials.

It is still an open problem to find a combinatorial interpretation of $K_{\lambda\mu}(q,t).$

Haglund's conjecture

In [Hag10], Jim Haglund conjectures that

\[ \left\langle \frac{\macdonaldJ_\lambda(\xvec;q,q^k)}{(1-q)^{|\lambda|}} , \schurS_\mu(\xvec) \right\rangle \in \setN[q] \]

for any $k \in 0,1,2,\dotsc.$ In [HHL05], it is proved that the expression $\frac{\macdonaldJ_\lambda(\xvec;q,q^k)}{(1-q)^{|\lambda|}}$ when expanded in the $\monomial_\mu$-basis has coefficients in $\setN[q].$ Haglunds conjecture states that the expression is actually Schur positive.

Arun Ram suggests that Haglund's conjecture can be made stronger: there is some type of positivity when $\macdonaldJ_\lambda(\xvec;q,q^k)$ is expanded in the $\macdonaldJ_\lambda(\xvec;q,q^{k-1})$-basis. Note that $\macdonaldJ_\lambda(\xvec;q,q)=\schurS_\lambda.$

Haglund's conjecture has been shown to be true in some special cases, see [Yoo12YOO15Bha22].

A quasisymmetric refinement of $\macdonaldJ_\lambda(\xvec;q,t)$ is introduced in [CHMMW22]. The quasisymmetric Macdonald polynomials $G_{\alpha}(\xvec;q,t),$ as a sum over certain non-symmetric Macdonald polynomials, such that the following properties hold:

  • $\macdonaldJ_\lambda(\xvec;q,t)$ is a positive sum of $G_{\alpha}(\xvec;q,t).$
  • $G_{\alpha}(\xvec;q,t)$ is quasisymmetric.
  • $G_{\alpha}(\xvec;0,0)$ is a quasisymmetric Schur function.
  • The $G_{\alpha}(\xvec;q,t)$ have a combinatorial HHL-type formula.

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