Alcove Walks and GKM Theory for Affine Flags
Document Type
Conference Proceeding
Role
Author
Publication
INdAM Series: Approximation Theory and Numerical Analysis Meet Algebra, Geometry, Topology
Publisher
Springer Singapore
Standard Number
9789819765072
Volume
60
First Page
289
Last Page
318
Publication Date
12-23-2024
Abstract
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.
Repository Citation
Milićević, E., Taipale, K. (2024). Alcove Walks and GKM Theory for Affine Flags. In: Lanini, M., Manni, C., Schenck, H. (eds) Approximation Theory and Numerical Analysis Meet Algebra, Geometry, Topology. INdAM 2022. Springer INdAM Series, vol 60. Springer, Singapore. https://doi.org/10.1007/978-981-97-6508-9_14
