nLab
Macdonald polynomial

Macdonald polynomials are a generalization of a Schur functions; they unify a theory of Hall-Littlewood and Jack polynomials. They form a family of orthogonal polynomials? which are symmetric functions in x 1,,x nx_1,\ldots,x_n with coefficients which are rational functions of two additional variables qq and tt.

Given a partition λ\lambda, one defines a shift operator T q,x iT_{q,x_i} which maps f=f(x 1,,x n)f = f(x_1,\ldots, x_n) to f(x 1,,x i1,qx i,x i+1,,x n)f(x_1,\ldots, x_{i-1}, q x_i, x_{i+1},\ldots,x_n) and the operators D rD_r, r=0,1,,nr = 0, 1, \ldots, n via

D i=t r(r1)2 I{1,,n},|I|=r iI,jItx ix jx ix j iIT q,x i, D_i = 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},

and the corresponding generating series D:= r=0 nD ru rD := \sum_{r=0}^n D_r u^r.

The Macdonald polynomial P λ(x;q,t)P_\lambda(x;q,t) is an eigenfunction of DD with the eigenvalue

i=1 n(1+ut niq λ i) \prod_{i=1}^n (1 + u t^{n-i} q^{\lambda_i})

In the case q=tq = t we get the Schur function P λ(x;t,t)=s λ(t)P_\lambda(x; t,t) = s_\lambda(t). Similarly, shifted Macdonald polynomials generalize shifted Schur functions.

  • I. G. Macdonald, Publ. I.R.M.A. 372 (S-20), 131-171, 1988.
  • I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford Math. Monographs, 2nd enlarged ed. 1995
  • wikipedia Macdonald polynomial
  • A. M. Garsia, C. Procesi, On certain graded S nS_n-modules and the qq-Kostka polynomials, Adv. Math. 94 (1992) 82-138
  • A. Okounkov, (Shifted) Macdonald polynomials: q-integral representation and combinatorial formula, Compositio Math. 112 (1998), 147–182. MR99h:05120, doi, BC-type interpolation Macdonald polynomials and binomial formula for Koornwinder polynomials, Transform. Groups 3 (1998) 181–207, MR99h:33061, Combinatorial formula for Macdonald polynomials and generic Macdonald polynomials, Transform. Groups 8 (2003), no. 3, 293–305, MR2004e:05202, doi
  • N. Bergeron, A. M. Garsia, On certain spaces of harmonic polynomials, in: Hypergeometric functions on domains of positivity, Jack polynomials, and applications (Tampa, FL, 1991), Contemp. Math. 138, 51–86 (Amer. Math. Soc. 1992)
  • A. Yu. Okounkov, 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, doi
  • G. Felder, L. Stevens, A. Varchenko, Modular transformations of the elliptic hypergeometric functions, Macdonald polynomials, and the shift operator, Moscow Math. J. 3, n. 2 (2003), 457-473, pdf, arXiv:math.QA/0203049, MR2025269
  • Mark Haiman, Hilbert schemes, polygraphs and the Macdonald positivity conjecture, J. Amer. Math. Soc. 14 (2001), no. 4, 941–1006, MR2002c:14008, doi; Macdonald polynomials and geometry, in: New perspectives in algebraic combinatorics (Berkeley, CA, 1996–97), 207–254, Math. Sci. Res. Inst. Publ. 38, Cambridge Univ. Press 1999, pdf
  • M. Haiman, Cherednik algebras, Macdonald polynomials and combinatorics, Proc. ICM, Madrid 2006, Vol. III, 843-872, djvu scan, author’s pdf
  • M. Haiman, A. Woo, Geometry of qq and q,tq,t-analogs in combinatorial enumeration, in: Geometric combinatorics, 207–248, IAS/Park City Math. Ser. 13, Amer. Math. Soc., Providence, RI, 2007, pdf, ps
  • A. M. Garsia, M. Haiman, A graded representation model for Macdonald’s polynomials, Proc. Nat. Acad. Sci. U.S.A. 90 (1993) 3607–3610, MR94b:05206, PNAS
  • A. M. Garsia, G. Tesler, Plethystic formulas for Macdonald q,tq, t-Kostka coefficients, Advances in Math. 123 (1996) 144–222, MR1420484; A. M. Garsia, J. Remmel, Plethystic formulas and positivity for q,tq,t-Kostka coefficients, Mathematical essays in honor of Gian-Carlo Rota (Cambridge, MA, 1996), 245–262, Progr. Math. 161, Birkhäuser 1998, MR99j:05189d
  • Friedrich Knop, Integrality of two variable Kostka functions, J. Reine Angew. Math. 482 (1997), 177–189, doi, MR99j:05189c
  • Siddhartha Sahi, Interpolation, integrality, and a generalization of Macdonald’s polynomials, Internat. Math. Res. Notices 1996, no. 10, 457–471, MR99j:05189b, doi
  • Anatol N. Kirillov, Masatoshi Noumi, Affine Hecke algebras and raising operators for Macdonald polynomials, Duke Math. J. 93 (1998), no. 1, 1–39, MR99j:05189a, doi
  • Anatol Kirillov Jr., Traces of intertwining operators and Macdonald’s polynomials, q-alg/9503012
  • Anton Gerasimov, Dimitri Lebedev, Sergey Oblezin, Baxter operator formalism for Macdonald polynomials. arxiv/1204.0926
  • Persi Diaconis, Arun Ram, A probabilistic interpretation of the Macdonald polynomials, arxiv/1007.4779

Revised on March 7, 2013 17:05:01 by Zoran Škoda (161.53.130.104)