The Combinatorics of Higher Dimensional Catalan Polynomials

The Combinatorics of Higher Dimensional Catalan Polynomials The Combinatorics of Higher Dimensional Catalan Polynomials Fishel, after 9 years in industry, has returned to academia at Arizona State University. This proposal contains two projects. The major project concerns higher dimensional analogs of the theory of Macdonald 2-parameter symmetric functions. It is joint work with Ian Grojnowski, of Cambridge University. The theory of Macdonald symmetric functions is one of the most active areas in enumerative combinatorics. There is a vast body of work, with exciting applications and interactions with representation theory, algebraic geometry, statistics, and even physics (the topological vertex of string theory). At the heart of this theory is the interpretation, due to Haiman, of Macdonald polynomials in terms of the algebraic geometry of families of points in the plane C2, that is of the Hilbert scheme. This led to the proof of the positivity of Macdonald Kostka polynomials and the n!-theorem via a deep analysis of the algebraic geometry involved, and it has led to the discovery of new combinatorial statistics and techniques, culminating in a purely combinatorial proof of Macdonald positivity. It is a rich and beautiful theory, where exciting discoveries are still being made. The PI, joint with Grojnowski, proposes to generalize this to higher dimensions. In order to avoid the algebraic geometric difficulties involved, they study here only an analog of a small piece of the usual 2- dimensional theory, the theory of the (q, t)-Catalan polynomial These polynomials have two quite different definitions, and the equality of the two is one of the highpoints of the 2-dimensional theory. One definition, which is a straightforward consequence of the geometry, is as an alternating sum of rational functions. From its form it is not even obvious that this is a polynomial, though in fact it has non-negative coefficients. The other definition evidently has this property it is a weighted sum over Dyck paths, classical combinatorial objects. The proposal details the Fishel and Grojnowskis program for defining the combinatorial d-dimensional analogs of the Catalan polynomial. It explains detailed computations of the higher dimensional algebraic geometry, and motivated by this, detailed computations with a family of poset structures on the 1-skeleton of the associahedron, a simplicial complex whose vertices are the Dyck paths. The PI expects that further study of these structures will lead to the combinatorial description of the geometric quantities, and to the generalization of much of the theory of Macdonald symmetric functions. The second project is joint with Nelson, a graduate student of the PI at ASU. They seek to count the number of maximal, saturated chains in the Tamari lattice. The questions leading to this collaboration arose from her work on the main project.