Buildings, spiders, and geometric Satake

Abstract

AbstractLet$G$be a simple algebraic group. Labelled trivalent graphs called webs can be used to produce invariants in tensor products of minuscule representations. For each web, we construct a configuration space of points in the affine Grassmannian. Via the geometric Satake correspondence, we relate these configuration spaces to the invariant vectors coming from webs. In the case of$G= \mathrm{SL} (3)$, non-elliptic webs yield a basis for the invariant spaces. The non-elliptic condition, which is equivalent to the condition that the dual diskoid of the web is$\mathrm{CAT} (0)$, is explained by the fact that affine buildings are$\mathrm{CAT} (0)$.