Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Amit Hazi

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.

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

Abstract

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.
Paper Structure (5 sections, 21 theorems, 82 equations)

This paper contains 5 sections, 21 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Principal ideals
  4. Existence and rotatability
  5. Applications to the Hecke category

Key Result

Theorem 1.1

The two-colored Jones--Wenzl projector $\mathrm{JW}_R(\prescript{}{s}{n})$ exists if and only if $\genfrac{[}{]}{0pt}{}{n}{k}_{s}$ is invertible in $R$ for each integer $0 \leq k \leq n$.

Theorems & Definitions (44)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 2.1
  • proof
  • Theorem 2.2: el-univcox
  • Proposition 2.3
  • Lemma 3.1: Quantum Bézout's identity
  • proof
  • ...and 34 more