The Springer fiber of a matrix $X$ is a subvariety of the flag variety consisting of the flags fixed by $X$. One of the classical examples of geometric representation theory shows that their cohomology admits a representation of the symmetric group (or the Weyl group, in general Lie type). Like Schubert varieties, the geometry of Springer fibers is deeply entwined with combinatorics. Unlike Schubert varieties, very little is known about even straightforward questions about this geometry. In this talk, we study the combinatorics and geometry of a particular family of Springer fibers that arise in combinatorics, representation theory, and knot theory. We give some results about how to partition these Springer fibers into cells that are encoded by a kind of graph called a web.
The geometry and combinatorics of 3-row Springer fibers*
Julianna Tymoczko, Smith College
2022 AWM Research Symposium
Geometric and Categorical Aspects of Representation Theory and Related Topics