Webs and canonical bases in degree two

TL;DR

This paper demonstrates that Lusztig's canonical basis for degree two Grassmannian coordinate rings can be represented using SL(k) web diagrams, linking web invariants to web immanants in plabic graphs.

Contribution

It establishes a new connection between Lusztig's canonical basis and SL(k) web diagrams, providing a combinatorial description for degree two Grassmannian invariants.

Findings

Lusztig's canonical basis corresponds to SL(k) web diagrams in degree two.

Every SL(2) web immanant of a plabic graph is an SL(k) web invariant.

The work bridges web diagram combinatorics with Grassmannian coordinate ring structures.

Abstract

We show that Lusztig's canonical basis for the degree two part of the Grassmannian coordinate ring is given by SL(k) web diagrams. Equivalently, we show that every SL(2) web immanant of a plabic graph for Gr(k,n) is an SL(k) web invariant.