A characterization of indecomposable web modules over Khovanov–Kuperberg algebras

Abstract

After shortly reviewing the construction of the Khovanov–Kuperberg algebras, we give a characterization of indecomposable web modules. It says that a web module is indecomposable if and only if one can deduce its indecomposability directly from the Kuperberg bracket (via a Schur lemma argument). The proof relies on the construction of idempotents given by explicit foams. These foams are encoded by combinatorial data called red graphs. The key point is to show that when the Schur lemma does not apply for a web $w$, an appropriate red graph for $w$ can be found.