Grant: Discovery Grants Program - Individual

The primary focus of this project is the general problem of decomposing (or factoring) a permutation into a product of others. Permutation factorizations arise in many interesting and diverse contexts; for instance, they are intimately connected with the representation theory of the symmetric group, combinatorial topology (viz. embeddings of graphs on surfaces), singularity theory, algebraic geometry, and physics. This research program will investigate the algebraic and combinatorial structure of permutation factorizations, with a focus on enumerative aspects. Counting minimal length "transitive" decompositions of permutations into transposition factors is of particular interest, since this problem is known to be equivalent to the enumeration of almost simple branched coverings of the sphere by the sphere. It has recently been linked to the intersection theory of the moduli space of curves. The goal is to understand the combinatorics underlying transitive factorizations, motivated by their compelling rich structure and the long term hope that a deeper understanding will shed light on the associated geometry. To this end, connections between factorizations and well known combinatorial objects such as trees and parking functions will be studied. From an algebraic viewpoint, permutation factorization is at the heart of the representation theory of the symmetric group. (Indeed, counting factorizations is equivalent to determining the structure constants of the central subalgebra of the group algebra.) The algebraic aspects of permutation factorizations (as well as factorizations in other Coxeter groups) will be explored.