Buildings, spiders, and geometric Satake

By Bruce Fontaine, Joel Kamnitzer, Greg Kuperberg. Compositio Mathematica, 2011

53 citations recorded by the source corpus.

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 $G= \mathrm{SL} (3)$, 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 $\mathrm{CAT} (0)$.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2c076137dc0b759222a475e5270720a7819c0e02