Schur positivity and labeled binary treesAlgebra & Discrete Mathematics
|Speaker:||Vasu Tewari, University of Washington|
|Start time:||Mon, Nov 21 2016, 4:10PM|
Gessel introduced a multivariate formal power series tracking the distribution of ascents and descents in labeled binary
trees. In addition to showing that it was a symmetric function, he conjectured it was Schur positive in fact.
In this talk, I will present two proofs of this conjecture, the first of which utilizes a variant of a beautiful
bijection of Préville-Ratelle and Viennot concerning extensions of Tamari lattices, while the other involves solving a
functional equation in noncommutative variables. I will subsequently discuss connections to hyperplane arrangements, in
particular the Linial arrangement, by demonstrating a hidden symmetric group action on the regions of the Linial
arrangement. This uses a bijection found recently by Bernardi. Additionally, I will discuss gamma-positivity of the
coefficients of a polynomial considered by Postnikov in his work on alternating trees. Finally, time permitting, I will discuss
connections with the representation theory of the 0-Hecke algebra.
This is joint work with Ira Gessel and Sean Griffin.