Exotic Springer fibers for orbits corresponding to one-row bipartitions

TL;DR

This paper investigates the geometry and topology of exotic Springer fibers associated with one-row bipartitions, providing explicit combinatorial descriptions, affine pavings, and cohomology ring structures, linking to algebraic and diagrammatic frameworks.

Contribution

It offers a detailed combinatorial and geometric analysis of exotic Springer fibers for one-row bipartitions, including explicit affine pavings and cohomology computations, connecting to diagrammatic algebra.

Findings

Explicit affine paving of exotic Springer fibers

Cohomology ring structure computed via CW-complex

Connection to one-boundary Temperley-Lieb algebra

Abstract

We study the geometry and topology of exotic Springer fibers for orbits corresponding to one-row bipartitions from an explicit, combinatorial point of view. This includes a detailed analysis of the structure of the irreducible components and their intersections as well as the construction of an explicit affine paving. Moreover, we compute the ring structure of cohomology by constructing a CW-complex homotopy equivalent to the exotic Springer fiber. This homotopy equivalent space admits an action of the type C Weyl group inducing Kato's original exotic Springer representation on cohomology. Our results are described in terms of the diagrammatics of the one-boundary Temperley-Lieb algebra (also known as the blob algebra). This provides a first step in generalizing the geometric versions of Khovanov's arc algebra to the exotic setting.