Georgia Institute of Technology College of Sciences

A Springer fiber is the set of complete flags in Cn which are fixed by a given nilpotent matrix. It is a fundamental object of study in geometric representation theory and algebraic combinatorics. The irreducible components of a Springer fiber are indexed by combinatorial objects called standard Young tableaux. It is an open problem to describe geometric properties of these components (such as their singular loci and cohomology classes) in terms of the combinatorics of tableaux. We initiate a new approach to this problem by characterizing which irreducible components are equal to Richardson varieties, which are comparatively much better understood. Another motivation comes from Lusztig's recent study of the cell decomposition of the totally nonnegative part of a Springer fiber into totally positive Richardson cells. This is joint work in progress with Martha Precup.