nLab Macdonald polynomial

Idea

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 $x_1,\ldots,x_n$ 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

$$D_r = t^{\frac{r(r-1)}{2}} \sum_{I\subset \{1,\ldots,n\}, |I| = r} \prod_{i\in I, j\notin I} \frac{t x_i-x_j}{x_i-x_j}\prod_{i\in I} T_{q, x_i},$$

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

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

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

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.

References

• Ian G. Macdonald, A New Class of Symmetric Functions, Publ. I.R.M.A. 372 (S-20) (1988) 131-171 [web]

• Ian G. Macdonald, Symmetric functions and Hall polynomials, Oxford Math. Monographs, 2nd enlarged ed. (1995) [pdf]

• Wikipedia, Macdonald polynomial

• A. M. Garsia, C. Procesi, On certain graded $S_n$-modules and the $q$-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 $q$ and $q,t$-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 $q, t$-Kostka coefficients, Advances in Math. 123 (1996) 144–222, MR1420484; A. M. Garsia, J. Remmel, Plethystic formulas and positivity for $q,t$-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

• Anton Khoroshkin, Highest weight categories and Macdonald polynomials, arxiv/1312.7053

• E. Carlsson, E. Gorsky, A. Mellit, The $\mathbf{A}_{q,t}$ algebra and parabolic flag Hilbert schemes arxiv/1710.01407; A. Garsia, A. Mellit, Five-term relation and Macdonald polynomials, arxiv/1604.08655; A. Mellit, Plethystic identities and mixed Hodge structures of character varieties, arxiv/1603.00193

• Wy. Chuang, D-E. Diaconescu, R. Donagi, T. Pantev, Parabolic refined invariants and Macdonald polynomials, Commun. Math. Phys. 335, 1323–1379 (2015) doi

A string theoretic derivation is given for the conjecture of Hausel, Letellier and Rodriguez-Villegas on the cohomology of character varieties with marked points. Their formula is identified with a refined BPS expansion in the stable pair theory of a local root stack, generalizing previous work of the first two authors in collaboration with Pan. Haiman’s geometric construction for Macdonald polynomials is shown to emerge naturally in this context via geometric engineering. In particular this yields a new conjectural relation between Macdonald polynomials and refined local orbifold curve counting invariants. The string theoretic approach also leads to a new spectral cover construction for parabolic Higgs bundles in terms of holomorphic symplectic orbifolds.

category: algebra, combinatorics