Schur Positivity and Recent Trends in Algebraic Combinatorics
Abstract.
Schur positivity of symmetric functions plays a crucial role in
many outstanding problems of algebraic combinatorics, with natural ties
with several other areas of mathematics and theoretical physics. After
recalling the necessary background for symmetric function calculations, I
will recall how one may derive interesting enumerative combinatorics
identities from symmetric function identities involving positive Schur
function expressions. Next, I will explain why such expressions are very
rare in general, but frequent in the right context. I will then go on with
a survey of recent history and of some of the important current open
problems regarding Schur positivity, for instance in rectangular Catalan
combinatorics. If time allows several new conjectures will be presented,
as well as a general framework that unifies them.