Mark Haiman: (q,t,u)-Catalan combinatorics and the Schiffmann algebra
Abstract
Some beautiful combinatorics discovered in the last decade or so revolves around
two-parameter $(q,t)$ analogs of the Catalan numbers. The $(q,t)$ Catalan numbers
and their various friends and relations come from an algebra of operators that act
on symmetric functions in the theory of Macdonald polynomials. Thanks to work of
Schiffmann--Vasserot and Feigin--Tsymbauliak it is now known that these operators
form a representation of Schiffmann's 'Elliptic Hall algebra.'
The $(q,t)$ Catalan objects have a representation theoretic interpretation that
naturally extends to more than two parameters $(q,t,u,...)$. Almost nothing
combinatorial is known about this extension, although there are some nice conjectures.
For three parameters $(q,t,u)$, some of the open questions go back 25 years.
It turns out that the Schiffmann algebra depends symmetrically on three parameters
$q,t,u$ restricted to satisfy $q t u = 1$. Using this I can make some predictions about $(q,t,u)$ Catalan objects under this specialization of the parameters.