Research Statement

My research has been centered around the combinatorics of representations of symmetric groups, along with connections to Schubert calculus and permutation combinatorics. The specific family of representations I have been investigating are the (generalized) Specht modules. For each finite subset D of N×N (a diagram), there is an associated Specht module SD, a representation of the symmetric group S|D|. When D is the Ferrers diagram of a partition λ, we recover the classical Specht modules Sλ. As λ ranges over partitions of n, these are exactly the irreducible representations of Sn (over a field of characteristic zero). One would like a combinatorial rule for decomposing SD into a direct sum of irreducibles Sλ for any diagram D. This is a difficult problem, and is open for general diagrams, but deep combinatorics appears in the cases which are understood. For example, when D is a skew shape, SD decomposes according to the Littlewood-Richardson rule, and its dimension is the number of standard tableaux of shape D. Another class of diagrams for which SD has interesting combinatorics are th