A homomorphism between Bott-Samelson bimodules

Abstract

In the previous paper, we defined a new category which categorifies the Hecke algebra. This is a generalization of the theory of Soergel bimodules. To prove theorems, the existences of certain homomorphisms between Bott-Samuelson bimodules are assumed. In this paper, we prove this assumption. We only assume the vanishing of certain two-colored quantum binomial coefficients.

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

“…Two-colored Jones-Wenzl projectors lie at the heart of the Elias-Williamson construction of the diagrammatic Hecke category [4]. Recently Abe has shown that there is a "bimodule-theoretic" category (a modification of the category of classical Soergel bimodules) which is equivalent to the diagrammatic Hecke category under certain assumptions [2,1]. An important consequence of Theorem B (which we discuss in the final section) is that these assumptions essentially always hold.…”

“…These elements (defined in (5) below) are bivariate polynomials in [2] s and [2] t which are analogous to ordinary quantum numbers. For an integer 0 ≤ k ≤ n the two-colored quantum binomial coefficient [1] s can also be shown to be an element of R. Our first main result is the two-colored analogue of the well-known existence theorem for ordinary Jones-Wenzl projectors.…”

“…Conversely, by Lemma 4.3 there is a a diagram whose coefficient in JW Q ( s n) is exactly n−1 Proof of Theorem B.The condition on quantum binomial coefficients is the same as [1, Assumption 1.1]. By[1, Proposition 3.4] this implies that the quantum binomial and similarly for t, we conclude that[n+1]s [k]s = [n+1]t [k]t = 0 for all integers 1 ≤ k ≤ n.Conversely, if the two-colored Jones-Wenzl projectors exist and are rotatable, then (13) combined with Theorem A and Proposition 4.7 show that n+1 k s and n+1 k t…”

Koszul duality for Coxeter groups

“…As the reader might have noticed, we have used the same notation as for some objects in D BS (h, W ). This should not lead to any confusion, because of the following result due to Abe (see [A2,Theorem 3.15]).…”

IntroSurvey of representation theory

“…There is an interesting variation [Abe19] of the category of Soergel bimodules, that in few words one could describe as "Soergel bimodules+a specified localization". This category (provided that a few technical conditions coming from dihedral groups are satisfied [Abe21]) is better suited for positive characteristic (although Soergel calculus, not needing that technical condition seems like the winner in the competition of Hecke categories in positive characteristic). For example, for Weyl groups it works in all characteristics.…”