Buildings, spiders, and geometric Satake

Abstract

Let 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= 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 CAT (0), is explained by the fact that affine buildings are CAT (0). © © The Author(s) 2013.