Cyclic sieving, skew Macdonald polynomials and Schur positivity

The Schur Expansion of Macdonald Polynomials

Building on Haglund’s combinatorial formula for the transformed Macdonald polynomials, we provide a purely combinatorial proof of Macdonald positivity using dual equivalence graphs and give a combinatorial formula for the coefficients in the Schur expansion.

Positivity Questions for Cylindric Skew Schur Functions

Recent work of A. Postnikov shows that cylindric skew Schur functions, which are a generalisation of skew Schur functions, have a strong connection with a problem of considerable current interest: that of finding a combinatorial proof of the non-negativity of the 3-point Gromov-Witten invariants. After explaining this motivation, we study cylindric skew Schur functions from the point of view of...

Schubert Polynomials and Skew Schur Functions

Maximal supports and Schur-positivity among connected skew shapes

The Schur-positivity order on skew shapes is defined by B ≤ A if the difference sA − sB is Schur-positive. It is an open problem to determine those connected skew shapes that are maximal with respect to this ordering. A strong necessary condition for the Schur-positivity of sA−sB is that the support of B is contained in that of A, where the support of B is defined to be the set of partitions λ ...

A positivity result in the theory of Macdonald polynomials.

We outline here a proof that a certain rational function C(n)(q, t), which has come to be known as the "q, t-Catalan," is in fact a polynomial with positive integer coefficients. This has been an open problem since 1994. Because C(n)(q, t) evaluates to the Catalan number at t = q = 1, it has also been an open problem to find a pair of statistics a, b on the collection (n) of Dyck paths Pi of le...