A Graphical Calculus for Induction and Restriction on Temperley-Lieb Modules

Abstract

We develop a graphical calculus for induction and restriction along the Temperley-Lieb tower at a generic parameter. The main object, a diagrammatic 2-category $\mathfrak{D}$, has one-step generators that model the usual induction and restriction bimodules and additional two-step generators that model the summands that the cup-cap idempotents cut out in the two-strand Temperley-Lieb algebra. We construct an incarnation 2-functor from $\mathfrak{D}$ to the 2-category of Temperley-Lieb bimodules and prove a basis theorem for all 2-morphism spaces; the basis elements are indexed by bridges, a class of paths generalizing Dyck paths. The basis theorem implies that the incarnation functor is locally full and faithful; after we pass to the additive Karoubi envelope, this functor becomes an equivalence onto the corresponding 2-category of bimodules. Thus the calculus gives a concrete diagrammatic model for the functorial representation theory of the Temperley-Lieb tower. We also compute the split Grothendieck ring of the associated monoidal category and its action on the Grothendieck group of Temperley-Lieb modules. Under this action, homogenized Chebyshev polynomials represent the classes of standard modules and yield a positive integral basis. In contrast with Heisenberg categorifications, idempotent completion leaves the Grothendieck ring unchanged.