2023/03/21 by Elizabeth Milićević, Milićević, Elizabeth, Kaisa Taipale +1
Computer Science · Engineering · #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Image Processing and 3D Reconstruction #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.2303.12170
openalex publication_date 2023/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We develop the GKM theory for the torus-equivariant cohomology of the affine flag variety using the combinatorics of alcove walks. Dual to the usual GKM setup, which depicts the orbits of the small torus action on a graph, alcove walks take place in tessellations of Euclidean space. Walks in affine rank two occur on triangulations of the plane, providing a more direct connection to splines used for approximating surfaces. Alcove walks in GKM theory also need not be minimal length, and can instead be randomly generated, giving rise to more flexible implementation. This work reinterprets and recovers classical results in GKM theory on the affine flag variety, generalizing them to both non-minimal and folded alcove walks, all motivated by applications to splines.