Symmetric functions, tilings, and the nabla operator
李宜霖博士國立台灣師範大學數學系博士後研究員 15:30–17:00 鴻經館M107Abstract：
The study of Macdonald symmetric polynomials has produced many interesting combinatorial objects. Perhaps the most famous and well-studied such objects are the $q,t$-Catalan numbers, which can be defined combinatorially as the sum over Dyck paths weighted by the area and dinv statistics. Numerous generalizations of the $q,t$-Catalan numbers have been developed, including extensions to Schroder paths and to nested families of Dyck paths. All of these objects have natural interpretations in terms of the nabla operator $\nabla$ on symmetric functions.
In this talk, I will introduce the algebraic and combinatorial background of the nabla operator and present its new connections with domino tilings of a certain region on the square lattice. In particular, a product formula for the $q,t$-generalization of domino tilings of the Aztec diamond, together with a combinatorial proof of the joint symmetry of the area and dinv statistics on the Aztec diamond, is presented. If time permits, I will also outline some proof ideas and related results on tiling enumeration. This talk does not
assume any prior background.