Séminaire Lotharingien de Combinatoire, 95B.81 (2026), 12 pp.
Christian Gaetz, Jessica Striker, Joshua P. Swanson and Haihan Wu
Web Bases, Pocket Complexes, and Affine Buildings
Abstract.
A major research program in combinatorial representation theory is to
produce effectively computable graphical calculi which fully encode
representation categories of interest. Recent work of
Gaetz-Pechenik-Pfannerer-Striker-Swanson introduced fully
reduced 4-row hourglass plabic graphs, which, together with work of
Cautis-Kamnitzer-Morrison, completes this program
for Uq(sl4). These objects come
with certain moves. Here we build on work of
Fontaine-Kamnitzer-Kuperberg and describe how each such move-class may
be assembled into a certain highly structured 3-dimensional simplicial complex we call a pocket.
One of the many remarkable properties of pockets is that they embed in the affine building Δ(SL(4)∨). This is a very complicated, uncountable simplicial complex built from the affine Grassmannian and intimately involved in the geometric Satake correspondence. Pockets provide finite models for the geometry of this building.
Furthermore, pockets possess SL(4)-weight-valued geodesics and are
CAT(0) spaces. Special cases include the well-known tetrahedral-octahedral honeycomb. Pockets are a source of distributive lattices, including alternating sign matrices, tilings of the Aztec diamond, plane partitions, and more.
Received: November 15, 2025.
Accepted: February 15, 2026.
Final version: April 1, 2026.
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