Existence and rotatability of the two-colored Jones-Wenzl projector

Abstract

The two-colored Temperley-Lieb algebra $2{\mathrm{TL}}_R({}_{s}n)$ is a generalization of the Temperley-Lieb algebra. The analogous two-colored Jones-Wenzl projector ${\mathrm{JW}}_R({}_{s}n)\in 2{\mathrm{TL}}_R({}_{s}n)$ plays an important role in the Elias-Williamson construction of the diagrammatic Hecke category. We give conditions for the existence and rotatability of ${\mathrm{JW}}_R({}_{s}n)$ in terms of the invertibility and vanishing of certain two-colored quantum binomial coefficients. As a consequence, we prove that Abe's category of Soergel bimodules is equivalent to the diagrammatic Hecke category in complete generality.

arXiv comment16 pages, many figures, best viewed in color

References in corpus (1)

• The two-color Soergel calculus

Cited by in corpus (1)

• Koszul duality for Coxeter groups