Buildings, spiders, and geometric Satake

TL;DR

This paper explores the connection between webs, configuration spaces, and the geometric Satake correspondence for simple algebraic groups, providing new bases for invariant spaces in the case of SL(3).

Contribution

It constructs configuration spaces for webs in the affine Grassmannian and relates them to invariant vectors via geometric Satake, especially characterizing non-elliptic webs as CAT(0) spaces.

Findings

Non-elliptic webs form a basis for invariant spaces in SL(3).

Web dual diskoids are CAT(0), linking web properties to affine building geometry.

Configuration spaces relate webs to invariant vectors through geometric Satake.

Abstract

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).