Rotation-invariant web bases from hourglass plabic graphs

TL;DR

This paper introduces hourglass plabic graphs to construct the first rotation-invariant web basis for $U_q( ext{sl}_4)$, combining combinatorics, crystal theory, and skein relations.

Contribution

It develops a new diagrammatic framework with growth rules and algorithms for basis webs, unifying previous rotation-invariant web bases.

Findings

First rotation-invariant $U_q( ext{sl}_4)$-web basis constructed.

Provides growth rules and skein-based algorithms for web basis expansion.

Unifies previous web bases within the hourglass plabic graph framework.

Abstract

Webs give a diagrammatic calculus for spaces of tensor invariants. We introduce hourglass plabic graphs as a new avatar of webs, and use these to give the first rotation-invariant $U_q(\mathfrak{sl}_4)$-web basis, a long-sought object. The characterization of our basis webs relies on the combinatorics of these new plabic graphs and associated configurations of a symmetrized six-vertex model. We give growth rules, based on a novel crystal-theoretic technique, for generating our basis webs from tableaux and we use skein relations to give an algorithm for expressing arbitrary webs in the basis. We also discuss how previously known rotation-invariant web bases can be unified in our framework of hourglass plabic graphs.