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2 results

  • 10 - Combinatorial Aspects of Macdonald and Related Polynomials

  • Edited by Tom H. Koornwinder, Universiteit van Amsterdam, Jasper V. Stokman, Universiteit van Amsterdam
  • Book:
    Encyclopedia of Special Functions: The Askey-Bateman Project
    Published online:
    30 September 2020
    Print publication:
    15 October 2020, pp 314-367

  • Summary

    This chapter contains a survey of known results and open problems connected to the combinatorics of (type A) Macdonald polynomials. Macdonald polynomials are symmetric functions in a set of variables which depend on two extra parameters q,t. They include most of the commonly studied bases for the ring of symmetric functions, such as Schur functions and Hall-Littlewood polynomials, as special cases. Macdonald polynomials have geometric interpretations which make them important to algebraic geometry and mathematical physics, and are also fundamental to the study of special functions. Their combinatorial properties are rather mysterious, although a lot of progress has been made on the type A case in the past 20 years, in conjunction with the study of the representation theory of the ring of diagonal coinvariants. This survey shows how to express Macdonald polynomials, and other important objects such as the bigraded Hilbert series of the diagonal coinvariant ring, in terms of popular combinatorial structures including tableaux, Dyck paths, and parking functions.

  • A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path

  • J. Haglund, J. Morse, M. Zabrocki
  • Journal:
    Canadian Journal of Mathematics / Volume 64 / Issue 4 / 01 August 2012
    Published online by Cambridge University Press:
    20 November 2018, pp. 822-844
    Print publication:
    01 August 2012
    • Article
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  • We introduce a $q,\,t$-enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory $\nabla $ operator applied to a Hall–Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the “shuffle conjecture” (Duke J. Math. 126 (2005), pp. 195 − 232) for $\nabla {{e}_{n}}\left[ X \right]$. We bring to light that certain generalized Hall–Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of $q,\,t$-Catalan sequences, and we prove a number of identities involving these functions.