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

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