The Combinatorics of Macdonald Polynomials

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The PI plans to work on three related projects involving the combinatoricsof Macdonald polynomials and the space of diagonal harmonics. The firstis an attemptto build on recent joint work between Haiman, Loehr and the PI which gives acombinatorial formula for type A nonsymmetric Macdonald polynomials.The existence of this formula in the type A setting suggests that similarformulas exist for other root systems; the discovery of such formulaswould open up the subject significantly, as very few explicit identities ofany kind have been discovered beyond the type A case. The second projectattempts to build onthe combinatorial formula for (type A) symmetric Macdonald polynomials,which was discovered empirically by the PI and proved in subsequent jointworkwith Haiman and Loehr. The main open problem in this regard is to finda combinatorial formula for the coefficients in theSchur expansion; as a step in this directionthe PI describes a new conjectured formula for the special case ofaugmented hook shapes.The third project involves trying to prove a conjecturedformula for the monomial expansion of thecharacter of the space of diagonal harmonics, due to Haiman, Loehr, Remmel,Ulyanov and the PI. This conjecture,which led the PI to the empirical formula for symmetricMacdonald polynomials mentioned above,has a number of implications to enumerative combinatorics andrepresentation theory.Orthogonal polynomials are families of polynomials which satisfy anorthogonality condition, typically meaning that the integral of anytwo different elements of the family,against some given weight function, is zero.The study of orthogonal polynomials is several centuries old,and they have quite a number of applications throughout mathematics andscience.Symmetric functions are polynomials in several variables which areinvariant under any permutation of the variables. They have a large numberof applications to several branches of mathematics such as representationtheory andthe roots of polynomial equations.In 1988 Macdonald introduced a new family of orthogonal polynomials,which depend on a set of variables X and two extra parameters q,t. They aresymmetric functions in the variables X, andcontain most useful symmetric functions as special cases.They were immediately recognized as being important to several area ofmathematicsincluding combinatorics and special functions.In 2000 Haiman proved they have a complicated interpretation in terms ofan advanced,abstract and theoretical branch ofmathematics known as algebraic geometry.Macdonald's construction is indirect however, and neither it norHaiman's interpretationgive a way of easily describing these polynomials.Recently the PI discovered, and with Haiman and Loehr proved, a simplecombinatorial formula for Macdonald's polynomials. The PI is trying toapply this result to answer some of the open questions surrounding them.Macdonald and Cherednik now have a much more general construction whichcontains almost allpreviously studied symmetric functions and orthogonal polynomials asspecial cases.The PI, in collaboration with Haiman, is trying to extend thesimple combinatorial formula to this more general construction.