Grant: Discovery Grants Program - Individual

My research is in the area of algebraic combinatorics. The heart of this program is Weyl's character formula, a fundamental relation in character theory. Introduced by Hermann Weyl in the 1920s, it has had many proofs, both algebraic and combinatorial. It has also had many generalizations: even the classical theory includes an array of similar formulae involving other characters, such as symplectic Schur functions or orthogonal Schur functions, for other root systems. A modern twist on Weyl's character formula--a well-known classical algebraic identity--involves adding an extra parameter to the identity, thus deforming it. The proofs of these deformed versions of Weyl's character formula, and proofs of related formulae for various root systems, involve symmetric functions, tableaux, determinants, and alternating sign matrices, and reveal much about the structure of these identities. New understanding of the structure builds a framework which in turn leads to new results that are important in a subdomain of number theory where they have immediate and direct application. The challenge is to deform the Weyl character formulae in interesting and exploitable ways, interpret the identities in terms of the right sort of tableaux for the given root system, and execute combinatorial proofs.***In order to construct this theory, I need a comprehensive combinatorial framework with an interpretation in terms of tableaux and a corresponding set of algorithms (e.g. jeu de taquin, lattice path constructions) and algebraic techniques (e.g. determinantal manipulation) to devise and prove these theorems. In terms of specific results I propose 1) a lattice path proof of a Bn deformed Weyl denominator identity (this would be a fully combinatorial proof of a hybrid algebro-combinatorial proof due to Hamel and King), 2) proofs of both the Cn and Dn deformed Weyl denominator identities (related to results of Brubaker and Schultz), 3) refined/weighted enumerations with respect to various statistics of the alternating sign matrices involved in the preceding results (a direction suggested to me by both Behrend and Okada) and 4) proofs of various conjectures from the literature (Friedburg and Zhang, Brubaker et al.).***My recent work caught the attention of a community of analytic number theorists working on similar results from a different angle, and invitations to their workshops and conferences has resulted in a mutually beneficial cross pollination, hence I expect them to be an important audience for my combinatorial framework and these results. The work also has connections to the celebrated Langlands' program in number theory, an ambitious and visionary set of conjectures from the late 1960s that has inspired investigation into connections between number theory and other areas of mathematics (e.g. harmonic analysis) and even physics, e.g. string theory and quantum field theory.**