Mirković-Vilonen Basis

The Mirković-Vilonen basis is a geometric basis of the coordinate ring of a maximal unipotent group in a connected complex semisimple algebraic group . Its elements are indexed by stable Mirković-Vilonen cycles in the affine Grassmannian for the Langlands dual group. Geometric Satake gives bases of highest-weight representations from the fundamental classes of the cycles. These assemble into the basis of (Baumann et al. 2021).

Kamnitzer (2023) conjectured that this basis equals the theta basis, which is constructed from scattering diagrams for the cluster algebra structure on (Gross et al. 2018). Tang (2026) reports equality for every such . Tang credits ChatGPT-6 Pro with developing the proof and repairing errors identified in earlier drafts by automated review. Tang reports that the revised proof passed Rethlas-OSV automated review.