Nabla operator and combinatorial aspects of Atiyah-Bott Lefschetz theorem.

摘要

This thesis consists of two different parts. In the first part, we give a proof of the q, t-square conjecture of Loehr and Warrington. The proof is a joint work with N. Loehr and J. Haglund.{09}The conjecture, in the spirit of the q, t-Catalan theorem; roughly states the equality between a certain symmetric function identity and a combinatorial sum, called the q, t-Square series. In the course of the proof of this conjecture we will exhibit many plethystic identities which are interesting to know in their own right. In the second part, we consider Hilbert scheme of points in the plane and related varieties, in particular, we study smooth nested Hilbert scheme of points. Our purpose is to compute an Atiyah-Bott Lefschetz type formula on the smooth nested Hilbert scheme of points. As a result, we obtain a family of combinatorially appealing series in variables q and t. On the way, we find a combinatorially indexed set of generators for tangent spaces at torus fixed points of the smooth nested Hilbert schemes. We also show that the zero fiber of this scheme is locally complete intersection. This work can be seen as a continuation of Haiman's computation of Atiyah-Bott formula on Hilbert scheme of points, which produces the q, t-Catalan series.