Combinatorics of Macdonald polynomials
Abstract: Macdonald polynomials are orthogonal polynomials associated to root systems and depend on parameters q and t. Many familiar families of polynomials can be obtained from the Macdonald polynomials by specializing the values of q and t. As an example in the Type A root system, setting q=0 gives the Hall-Littlewood polynomials, and setting q=t=0 gives the Schur functions.
Macdonald polynomials can be constructed by multiplying intertwining operators of the double affine Hecke algebra. Using objects known as alcove walks, we give two combinatorial formulas; one for the expansion of a Macdonald polynomial in terms of monomials, and one for the expansion of the product of two Macdonald polynomials in the Macdonald basis.