2023-10-27
Macdonald P polynomials
Macdonald polynomials, $\{\macdonaldP_\lambda(\xvec;q,t) \}_{\lambda}$ 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. Colmenarejo–Ram suggests the term bosonic Macdonald polynomials while the non-symmetric Macdonald polynomials are called electronic Macdonald polynomials [CR22b].
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.
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
\[ b_{\mu}(s;q,t) \coloneqq \begin{cases} \frac{ 1-q^{\arm_\mu(s)}t^{1+\leg_\mu(s)} }{ 1-q^{1+\arm_\mu(s)}t^{\leg_\mu(s)} } &\text{if $s\in \mu$} \\ 1 &\text{otherwise}. \end{cases} \]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 $\macdonaldE^{id}_{\gamma}(x_1,\dotsc,x_n;q,t) = \macdonaldE_{\gamma}(x_n,\dotsc,x_1;q^{-1},t^{-1}).$
Another similar expression ([CR22b]) is
\begin{equation*} \macdonaldP_\lambda(\xvec;q,t) = \frac{1}{W_\lambda(t)} \sum_{w \in \symS_n} w \left( \macdonaldE_{\gamma}(\xvec;q,t) \prod_{i \lt j} \frac{x_i - t x_j}{x_i - x_j} \right), \end{equation*}where $W_\lambda(t)$ is a normalization constant, so that $[x^{\lambda}] \macdonaldP_\lambda =1.$ Compare this with the formula for Hall–Littlewood polynomials (obtained as $q\to 0$).
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].
Quasisymmetric Macdonald J polynomials
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 $\setN[q,t].$
Wreath Macdonald P polynomials
The wreath Macdonald P polynomials, $\macdonaldP_{\lambdavec^\bullet}(\xvec;q,t),$ were conjectured to exist in M. Haiman's seminal article [7.2.19, Hai02a]. This conjecture was later proved in [BFV14]. See [OS23] for a survey on wreath Macdonald P polynomials.
References
- [Ale19a] Per Alexandersson. Non-symmetric Macdonald polynomials and Demazure–Lusztig operators. Séminaire Lotharingien de Combinatoire, 76, 2019.
- [BFV14] Roman Bezrukavnikov, Michael Finkelberg and Vadim Vologodsky. Wreath Macdonald polynomials and the categorical MCKAY correspondence. Cambridge Journal of Mathematics, 2(2):163–190, 2014.
- [Bha22] Aritra Bhattacharya. Haglund's positivity conjecture for multiplicity one pairs. arXiv e-prints, 2022.
- [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.
- [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.
- [CR22b] Laura Colmenarejo and Arun Ram. c-functions and Macdonald polynomials. arXiv e-prints, 2022.
- [GL19b] Camilo González and Luc Lapointe. Multi-Macdonald polynomials. arXiv e-prints, 2019.
- [Hag10] Jim Haglund. Combinatorics associated to type $a$ nonsymmetric Macdonald polynomials. Oberwolfach report 2010.
- [Hai01] Mark D. Haiman. Hilbert schemes, polygraphs and the Macdonald positivity conjecture. Journal of the American Mathematical Society, 14(04):941–1007, October 2001.
- [Hai02a] Mark Haiman. Combinatorics, symmetric functions and Hilbert schemes. Current developments in mathematics, 2002(1):39–111, 2002.
- [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.
- [HHL08] James Haglund, Mark D. Haiman and Nicholas A. Loehr. A combinatorial formula for nonsymmetric Macdonald polynomials. American Journal of Mathematics, 130(2):359–383, 2008.
- [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
- [OS23] Daniel Orr and Mark Shimozono. Wreath Macdonald polynomials, a survey. arXiv e-prints, 2023.
- [Yoo12] Meesue Yoo. A combinatorial formula for the schur coefficients of the integral form of the Macdonald polynomials in the two column and certain hook cases. Annals of Combinatorics, 16(2):389–410, February 2012.
- [YOO15] Meesue YOO. Schur coefficients of the integral form Macdonald polynomials. Tokyo Journal of Mathematics, 38(1), June 2015.