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

TL;DR

This paper investigates the conditions under which the two-colored Jones-Wenzl projector exists and can be rotated, linking algebraic properties to the structure of the diagrammatic Hecke category.

Contribution

It provides new criteria for the existence and rotatability of the two-colored Jones-Wenzl projector based on quantum binomial coefficients, and establishes an equivalence between Abe's bimodule category and the diagrammatic Hecke category.

Findings

Criteria for existence and rotatability based on quantum binomial coefficients

Proved equivalence of Abe's category and the diagrammatic Hecke category

General conditions applicable to the two-colored Temperley-Lieb algebra

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.