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

Abstract

The two-colored Temperley-Lieb algebra 2TL R ( s n) is a generalization of the Temperley-Lieb algebra. The analogous two-colored Jones-Wenzl projector JW R ( s n) ∈ 2TL 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 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.

On Frobenius exact symmetric tensor categories

“…We start by observing that under condition (1), homotopical Nboundedness is characterised as in the last phrase of the theorem. Indeed, by construction and the defining property of rotatability in [Ha,§2], both conditions in Corollary 3.1.4 become equivalent under assumption (1).…”

“…By Proposition 1.4.7(3) there exists an l ≤ n that does not divide m with [[l]] R = 0. By maximality of m we also know that m does not divide l. But [Ha,Lemma 3.4] then implies that [[m]] R and [[l]] R cannot both be zero, a contradiction. The identical argument for R ′ concludes the proof.…”

“…then these relations determine them uniquely. Indeed, the polynomials ν i (x, y) exist, see [Ha,Lemma 3.2]. For example, ν N −1 (x, x) is given by the product in Lemma 1.3.2(2), where j is now restricted by the condition gcd(j, N ) = 1.…”

“…The equivalence of (1) and ( 2) is [Ha, Theorem B]. The equivalence of (1) and ( 3) is a reformulation of [Ha,Proposition 4.7], using the proof of [Ha,Lemma 4.6].…”

Koszul duality for Coxeter groups

“…[EW3,§7]), but we will not consider this question here. Assumptions ( 2) and ( 3) are also necessary for the theory of [EW2], hence also for all of its applications, although this was not made explicit before [EW3,Ha]. (In particular, they should be imposed in [AMRW1] and in [ARV].)…”