Let G be a simple algebraic group. Labelled trivalent graphs called webs can
be used to product 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 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).Comment: 49 pages; revised and to appear in Compositio Mathematic
