The q,t-Catalan Numbers and the Space of Diagonal Harmonics - with an Appendix on the Combinatorics of Macdonald Polynomials
Macdonald polynomials form a distinguished family of orthogonal polynomials in several variables. Properties of those polynomials, combinatorics connected with them and a circle of ideas around them were developed systematically in the last few decades, together with various interesting applications. This is the main topic of the book, which is based on lectures given by the author at the University of Pennsylvania. The book starts with a summary of the properties of q-analogues of counting functions and symmetric functions. Macdonald polynomials and diagonal harmonics, together with an historical background are treated next. The core of the book starts with a discussion of the properties of the q,t-Catalan numbers and q,t-Schröder numbers. The main topic is then the so-called “shuffle” conjecture, which is presented together with its proof in the case of hook shapes. The book contains a lot of exercises (with solutions of many of them in appendix C).