Combinatorics Seminar: $(q,t,u)$-Catalan combinatorics and the Schiffmann algebra

Seminar | September 30 | 12:10-1 p.m. | 939 Evans Hall

 Mark Haiman, UC Berkeley

 Department of Mathematics

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.