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\section*{Macdonald polynomial}

\textbf{Macdonald polynomials} are a generalization of a [[Schur function]]s; they unify a theory of Hall-Littlewood and [[Jack polynomial]]s. They form a family of [[orthogonal polynomials]] which are [[symmetric functions]] in $x_1,\ldots,x_n$ with coefficients which are rational functions of two additional variables $q$ and $t$.

Given a [[partition]] $\lambda$, one defines a shift operator $T_{q,x_i}$ which maps $f = f(x_1,\ldots, x_n)$ to $f(x_1,\ldots, x_{i-1}, q x_i, x_{i+1},\ldots,x_n)$ and the operators $D_r$, $r = 0, 1, \ldots, n$ via

\begin{displaymath}
D_r = t^{\frac{r(r-1)}{2}} \sum_{I\subset \{1,\ldots,n\}, |I| = r} \prod_{i\in I, j\notin I} \frac{t x_i-x_j}{x_i-x_j}\prod_{i\in I} T_{q, x_i},
\end{displaymath}
and the corresponding generating series $D := \sum_{r=0}^n D_r u^r$.

The \textbf{Macdonald polynomial} $P_\lambda(x;q,t)$ is an eigenfunction of $D$ with the eigenvalue

\begin{displaymath}
\prod_{i=1}^n (1 + u t^{n-i} q^{\lambda_i})
\end{displaymath}
In the case $q = t$ we get the [[Schur function]] $P_\lambda(x; t,t) = s_\lambda(t)$. Similarly, shifted Macdonald polynomials generalize shifted Schur functions.

\begin{itemize}%
\item I. G. Macdonald, Publ. I.R.M.A. 372 (S-20), 131-171, 1988.
\item I. G. Macdonald, \emph{Symmetric functions and Hall polynomials}, Oxford Math. Monographs, 2nd enlarged ed. 1995
\item wikipedia \href{http://en.wikipedia.org/wiki/Macdonald_polynomials}{Macdonald polynomial}
\item A. M. Garsia, C. Procesi, \emph{On certain graded $S_n$-modules and the $q$-Kostka polynomials}, Adv. Math. \textbf{94} (1992) 82-138
\item A. Okounkov, \emph{(Shifted) Macdonald polynomials: q-integral representation and combinatorial formula}, Compositio Math. \textbf{112} (1998), 147--182. \href{http://www.ams.org/mathscinet-getitem?mr=1626029}{MR99h:05120}, \href{http://dx.doi.org/10.1023/A:1000436921311}{doi}, \emph{BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials}, Transform. Groups \textbf{3} (1998) 181--207, \href{http://www.ams.org/mathscinet-getitem?mr=1628453}{MR99h:33061}, \emph{Combinatorial formula for Macdonald polynomials and generic Macdonald polynomials}, Transform. Groups \textbf{8} (2003), no. 3, 293--305, \href{http://www.ams.org/mathscinet-getitem?mr=1996418}{MR2004e:05202}, \href{http://dx.doi.org/10.1007/s00031-003-0306-0}{doi}
\item N. Bergeron, A. M. Garsia, \emph{On certain spaces of harmonic polynomials}, in: Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991), Contemp. Math. \textbf{138}, 51--86 (Amer. Math. Soc. 1992)
\item A. Yu. Okounkov, \emph{A remark on the Fourier pairing and the binomial formula for the Macdonald polynomials}, Funktsional. Anal. i Prilozhen. 36 (2002), no. 2, 62--68, 96; translation in Funct. Anal. Appl. 36 (2002), no. 2, 134--139, \href{http://dx.doi.org/10.1023/A:1015670523770}{doi}
\item G. Felder, L. Stevens, [[A. Varchenko]], \emph{Modular transformations of the elliptic hypergeometric functions, Macdonald polynomials, and the shift operator}, Moscow Math. J. \textbf{3}, n. 2 (2003), 457-473, \href{http://www.ams.org/distribution/mmj/vol3-2-2003/felder-etal.pdf}{pdf}, \href{http://lanl.arxiv.org/abs/math/0203049}{arXiv:math.QA/0203049}, \href{http://www.ams.org/mathscinet-getitem?mr=2025269}{MR2025269}
\item Mark Haiman, \emph{Hilbert schemes, polygraphs and the Macdonald positivity conjecture}, J. Amer. Math. Soc. \textbf{14} (2001), no. 4, 941--1006, \href{http://www.ams.org/mathscinet-getitem?mr=1839919}{MR2002c:14008}, \href{http://dx.doi.org/10.1090/S0894-0347-01-00373-3}{doi}; \emph{Macdonald polynomials and geometry}, in: New perspectives in algebraic combinatorics (Berkeley, CA, 1996--97), 207--254, Math. Sci. Res. Inst. Publ. \textbf{38}, Cambridge Univ. Press 1999, \href{http://math.berkeley.edu/~mhaiman/ftp/nfact/msri.pdf}{pdf}
\item M. Haiman, \emph{Cherednik algebras, Macdonald polynomials and combinatorics}, Proc. ICM, Madrid 2006, Vol. III, 843-872, \href{http://www.mathunion.org/ICM/ICM2006.3/Main/icm2006.3.0843.0872.ocr.djvu}{djvu scan}, \href{http://math.berkeley.edu/~mhaiman/ftp/icm-2006/comb-mac-chered.pdf}{author's pdf}
\item M. Haiman, A. Woo, \emph{Geometry of $q$ and $q,t$-analogs in combinatorial enumeration}, in: Geometric combinatorics, 207--248, IAS/Park City Math. Ser. \textbf{13}, Amer. Math. Soc., Providence, RI, 2007, \href{http://math.berkeley.edu/~mhaiman/ftp/pcmi-2004/notes.pdf}{pdf}, \href{http://math.berkeley.edu/~mhaiman/ftp/pcmi-2004/notes.ps}{ps}
\item A. M. Garsia, M. Haiman, \emph{A graded representation model for Macdonald's polynomials}, Proc. Nat. Acad. Sci. U.S.A. \textbf{90} (1993) 3607--3610, \href{http://www.ams.org/mathscinet-getitem?mr=1214091}{MR94b:05206}, \href{http://www.pnas.org/cgi/reprintframed/90/8/3607}{PNAS}
\item A. M. Garsia, G. Tesler, \emph{Plethystic formulas for Macdonald $q, t$-Kostka coefficients}, Advances in Math. \textbf{123} (1996) 144--222, \href{http://www.ams.org/mathscinet-getitem?mr=1420484}{MR1420484}; A. M. Garsia, J. Remmel, \emph{Plethystic formulas and positivity for $q,t$-Kostka coefficients}, Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996), 245--262, Progr. Math. \textbf{161}, Birkh\"a{}user 1998, \href{http://www.ams.org/mathscinet-getitem?mr=1627327}{MR99j:05189d}
\item Friedrich Knop, \emph{Integrality of two variable Kostka functions}, J. Reine Angew. Math. 482 (1997), 177--189, \href{http://dx.doi.org/10.1515/crll.1997.482.177}{doi}, \href{http://www.ams.org/mathscinet-getitem?mr=1427661}{MR99j:05189c}
\item Siddhartha Sahi, \emph{Interpolation, integrality, and a generalization of Macdonald's polynomials}, Internat. Math. Res. Notices 1996, no. 10, 457--471, \href{http://www.ams.org/mathscinet-getitem?mr=1399411}{MR99j:05189b}, \href{http://dx.doi.org/10.1155/S107379289600030X}{doi}
\item Anatol N. Kirillov, Masatoshi Noumi, \emph{Affine Hecke algebras and raising operators for Macdonald polynomials}, Duke Math. J. \textbf{93} (1998), no. 1, 1--39, \href{http://www.ams.org/mathscinet-getitem?mr=1620075}{MR99j:05189a}, \href{http://dx.doi.org/10.1215/S0012-7094-98-09301-2}{doi}
\item Anatol Kirillov Jr., \emph{Traces of intertwining operators and Macdonald's polynomials}, \href{http://arxiv.org/abs/q-alg/9503012}{q-alg/9503012}
\item Anton Gerasimov, Dimitri Lebedev, Sergey Oblezin, \emph{Baxter operator formalism for Macdonald polynomials}. \href{http://arxiv.org/abs/1204.0926}{arxiv/1204.0926}
\item Persi Diaconis, [[Arun Ram]], \emph{A probabilistic interpretation of the Macdonald polynomials}, \href{http://arxiv.org/abs/1007.4779}{arxiv/1007.4779}
\item Anton Khoroshkin, \emph{Highest weight categories and Macdonald polynomials}, \href{http://arxiv.org/abs/1312.7053}{arxiv/1312.7053}
\item E. Carlsson, E. Gorsky, A. Mellit, The $\mathbf{A}_{q,t}$ algebra and parabolic flag Hilbert schemes \href{https://arxiv.org/abs/1710.01407}{arxiv/1710.01407}; A. Garsia, A. Mellit, \emph{Five-term relation and Macdonald polynomials}, \href{https://arxiv.org/abs/1604.08655}{arxiv/1604.08655}; A. Mellit, \emph{Plethystic identities and mixed Hodge structures of character varieties}, \href{https://arxiv.org/abs/1603.00193}{arxiv/1603.00193}

\end{itemize}
category: algebra, combinatorics [[!redirects Macdonald polynomials]]



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