nLab Macdonald polynomial

Contents

Contents

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

References

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  • Wikipedia, Macdonald polynomial

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  • 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

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  • 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

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  • 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

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

  • E. Carlsson, E. Gorsky, A. Mellit, The A q,t\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.

Last revised on January 15, 2023 at 18:05:19. See the history of this page for a list of all contributions to it.