In this project, Warrington will explore three families of polynomials central to the field of algebraic combinatorics, each lying at the intersection of representation theory, geometry and combinatorics: (1) The Kazhdan-Lusztig (KL) polynomials comprise the first family. Here, Warrington will study the combinatorics of the mu-coefficients of KL polynomials through the so-called 'crosshatch pairs,' introduced by the PI. Central to these investigations will be the computer software (in ongoing development) that he is writing and which will be made publically available. (2) Plethysm is one of three basic ways of 'multiplying' irreducible characters of the symmetric group. The Frobenius map provides a correspondence between irreducible characters of the symmetric group and the Schur functions in the ring of symmetric functions. In light of this, it is often desirable to be able to expand a given symmetric function in terms of Schur functions. This is especially desirable when working with plethysms. However, combinatorial Schur expansions are notoriously difficult to describe. Along with N. Loehr, Warrington intends to continue his intermediate goal of finding expansions in terms of Gessel's fundamental quasisymmetric functions. (3) q, t-Catalan numbers. The q, t-Catalan numbers and related polynomials illustrate rich connections between the diagonal harmonics module and the combinatorics of weighted lattice paths. Warrington intends to respond in the obvious ways to the fact that conjectures are outpacing theorems in this arena. He will develop new combinatorial tools to directly prove many of the outstanding conjectures.
The symmetric group of permutations is a fundamental object in mathematics. It arises from symmetry and appears with a corresponding ubiquity in mathematics and our world. It plays roles in cryptography, in our theories of quantum chemistry and in such mundane acts as the shuffling of cards. One of the most fruitful ways of probing the algebraic structure of the symmetric group is through representation theory. Representations are often associated with polynomials whose coefficients count things. Understanding the combinatorial underpinnings, that is, determining precisely what objects are being counted, illuminates the corresponding algebra and geometry. Additionally, the combinatorics of these polynomials is exceedingly rich; the ensuing combinatorics is worth studying for its own sake.
