The Chromatic Category: A Connection Between Planar Graph Colouring and Representation Theory

Abstract

The chromatic category 𝒞(δ) is a diagrammatic monoidal category that encodes information about the proper colourings of the duals of planar graphs. In this work, we show that the chromatic category is pivotal, provide a basis for its morphisms spaces, and show that it is related to several other categories in the literature. In particular, under some assumptions, we show that 𝒞(δ) is isomorphic to certain monoidal subcategories of 𝖘𝖑₂-mod and U_q(𝖘𝖑₂)-mod, monoidally generated by the irreducible 3-dimensional representation in both cases. We show 𝒞(δ) exhibits a close relationship to the Temperley-Lieb category, which is also isomorphic to a category of representations (instead monoidally generated by the 2-dimensional irreducible representations of 𝖘𝖑₂ and U_q(𝖘𝖑₂). We also demonstrate that 𝒞(δ) has some relation to the Kauffman skein category. We extend many results of Fendley and Krushkal from the chromatic algebra to the chromatic category.