Combinatorial and algebraic methods in Schubert geometry

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The PI will study combinatorial, algebraic and computational questions about the geometry of Schubert varieties in Grassmannians and flag manifolds. The two main projects explore central problems in the subject. The first project seeks uniformly stated combinatorial rules in Schubert calculus, via an expanded theory of Young tableaux. The famous Littlewood-Richardson coefficients arise in this context, but also in the theory of representations of general linear groups and of symmetric groups, eigenvalues of sums of Hermitian matrices and short exact sequences of finite abelian p-groups. The PI will find combinatorial rules for these coefficients and their Schubert calculus extensions, building on joint work with H. Thomas. The second project aims to further develop a combinatorial and computational framework, introduced jointly with A. Woo, to understand the singularities of Schubert varieties. The main tools applied and further developed in this project come from combinatorics, including algebraic and geometric combinatorics, and combinatorial commutative algebra. Combinatorics concerns discrete objects such as permutations, partial orders and graphs, as well as techniques of enumeration. In many instances, such as with Schubert varieties, continuous objects can be parameterized by discrete data, opening the door for combinatorial analysis. Moreover, one often witnesses that the same combinatorial objects govern a priori different mathematical settings. The Littlewood-Richardson coefficients are a prime example of this. The project seeks to similarly find further cross-flow of ideas between areas of mathematics, as well as with other scientific disciplines.