Macdonald Polynomials and the Delta Conjecture

Since their introduction in 1988, Macdonald polynomials have played a central role in algebraic combinatorics. These polynomials are actually symmetric functions with coefficients in the field of rational functions in two variables q and t. Immediately after their introduction, a slightly modified version of the Macdonald polynomials has been conjectured to be Schur positive, i.e. to be a linear combination of Schur functions times certain polynomials with non-negative integer coefficients. Motivated by this conjecture, in the 90's Garsia and Haiman introduced the module of diagonal harmonics, i.e. the coinvariants of the diagonal action of the symmetric group on polynomials in two sets of n variables, and they conjectured that its Frobenius characteristic was given by a certain operator on symmetric functions for which the Macdonald polynomials form an eigenbasis. In 2001 Haiman proved the famous n! conjecture, (now n! theorem) and in 2002 he showed how this results implies the the result conjectured by Garsia and himself. Later on, Haglund, Haiman, Loehr, Remmel, and Ulyanov formulated the so called shuffle conjecture, i.e. they predicted a combinatorial interpretation for the same expression in terms of labelled Dyck paths, which refines the famous q,t-Catalan formulated by Haglund and then proved by Garsia and Haglund a few years before. This result was then proved in 2015 by Carlsson and Mellit, using a refinement by Haglund, Morse and Zabrocki. In the same year, Haglund, Remmel and Wilson conjectured a combinatorial formula for a slightly result, which they called \emph{Delta conjecture}, after the so called Delta operators, which they interpret combinatorially in terms of decorated labelled Dyck paths. In fact in the same article the authors conjectured a slightly more general combinatorial formula in terms of decorated partially labelled Dyck paths, which we call generalised Delta conjecture. One of the two results that we will present in this thesis is an important special case of the generalised Delta conjecture, the Schröder case, proved by D'Adderio, Vanden Wyngaerd, and myself. In 2007, Loehr and Warrington conjectured a combinatorial formula for yet another operator in terms of labelled square paths (ending East), called square conjecture. The Catalan case of this conjecture, known as q,t-square, has been proved earlier by Can and Loehr. The full square conjecture has been proved by Sergel in 2016 after the breakthrough of Carlsson and Mellit. The other main result that we will state in this thesis is the Schröder case of a new conjecture of D'Adderio, Vanden Wyngaerd, and myself, which extends the square conjecture of Loehr and Warrington in terms of decorated partially labelled square paths, that we call generalised Delta square conjecture. This conjecture extends the generalised Delta conjecture in the sense that on decorated partially labelled Dyck paths it gives the same combinatorial statistics. Some of the results regarding the Delta conjectures, expression by which we mean all the formulations and variants provided so far, are also related to parallelogram polyominoes. These objects and their relevant statistics will also be presented in this thesis, as they provide interesting material to understand the combinatorics behind the Delta operators. The combinatorics of lattice paths and parallelogram polyominoes, and the proofs of the Schröder cases of the two aforementioned conjectures, will constitute the body of this thesis