Oberseminar Sommersemester 2009

Leibniz Universitšt Hannover

Oberseminar zur Algebra und Algebraischen Kombinatorik

Montag, 4. und 11. und 18.5.2009, in Raum A 410

The combinatorics of Macdonald polynomials

Prof. Dr. Stephanie van Willigenburg (Vancouver/Hannover)

Macdonald polynomials were first introduced in 1988 in order to give a q-analogue of Selberg's integral. Since then, they have been related to many other areas including permutation statistics and diagonal harmonics. However, their abstract nature makes them challenging to work with. In this minicourse we will survey some of the combinatorial tools that have been discovered to make the manipulation of these polynomials more accessible, and survey some areas that these polynomials are newly impacting. Each talk will be accessible independently from the others.

Talk I

In this talk we will briefly overview the original definition of of Macdonald polynomials, and see how they relate to Schur polynomials, Zonal polynomials, Jack polynomials and Hall-Littlewood polynomials. We will then discuss combinatorial formulae for the integral form Macdonald polynomials and modified Macdonald polynomials, involving the attacking cells of Haglund, Haiman and Loehr. From here we will use our knowledge to obtain the (q,t)-Catalan sequence in terms of the combinatorial statistics, known as area and bounce, of Garsia and Haglund.

Talk II

Macdonald polynomials are symmetric polynomials involving parameters q,t. However, there also exists a non-symmetric analogue, known as the non-symmetric Macdonald polynomials. They arise naturally in other areas such as being the eigenfunctions of Cherednik algebras. These non-symmetric polynomials have an integral form, which can be described using the combinatorics of triple and attacking cells, and it is this definition that we will describe. From here, we will set the parameters q=t=0 to obtain Demazure atoms, and discuss their combinatorial properties, including a bijection involving semistandard Young tableaux, and an analogue of the Robinson-Schensted-Knuth algorithm, discovered by Mason.

Talk III

The most famous basis of the Hopf algebra of symmetric functions is the basis of Schur functions. One way to describe a Schur function is as a linear combination of certain Demazure atoms. Refining this linear combination we arrive at quasisymmetric Schur functions, which naturally partition Schur functions, and form a basis for the Hopf algebra of quasisymmetric functions. In this talk we will describe the Hopf algebraic properties of quasisymmetric functions and its dual algebra, using classical combinatorial constructs, such as descent sets of permutations, arising from the work of Gessel, Ehrenborg, and Gel'fand amongst others. Focussing on the basis of quasisymmetric Schur functions, we will conclude with some new combinatorial properties of these functions, and further avenues to pursue.