Existence and rotatability of the two‐colored Jones–Wenzl projector

The two‐colored Temperley–Lieb algebra is a generalization of the Temperley–Lieb algebra. The analogous two‐colored Jones–Wenzl projector plays an important role in the Elias–Williamson construction of the diagrammatic Hecke category. We give conditions for the existence and rotatability of 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.