Clasped web bases from hourglass plabic graphs

TL;DR

This paper extends the web basis construction from $U_q(rak{sl}_3)$ to $U_q(rak{sl}_4)$ using hourglass plabic graphs, providing combinatorial characterizations and establishing basis properties for tensor invariants.

Contribution

It introduces combinatorial characterizations of basis webs in the kernel of projection maps for $U_q(rak{sl}_4)$ tensor invariants, generalizing previous work for $U_q(rak{sl}_3)$.

Findings

Basis webs form a basis for tensor invariants.

Provides combinatorial criteria for basis webs.

Establishes clasped web bases for tensor products.

Abstract

G.-Pechenik-Pfannerer-Striker-Swanson applied hourglass plabic graphs to construct web bases for spaces of tensor invariants of fundamental representations of $U_q(\mathfrak{sl}_4)$, extending Kuperberg's celebrated basis for $U_q(\mathfrak{sl}_3)$. We give several combinatorial characterizations of basis webs in the kernel of the projection to invariants in a tensor product of arbitrary (type $1$) irreducibles. We apply this to show that the nonzero images of basis webs form a basis (a property shared with Lusztig's dual canonical basis) yielding distinguished clasped web bases for each such tensor product.