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.