Lifts of longest elements to braid groups acting on derived categories

Abstract

If we have a braid group acting on a derived category by spherical twists, how does a lift of the longest element of the symmetric group act? We give an answer to this question, using periodic twists, for the derived category of modules over a symmetric algebra. The question has already been answered by Rouquier and Zimmermann in a special case. We prove a lifting theorem for periodic twists, which allows us to apply their answer to the general case.Along the way we study tensor products in derived categories of bimodules. We also use the lifting theorem to give new proofs of two known results: the existence of braid relations and, using the theory of almost Koszul duality due to Brenner, Butler, and King, the result of Rouquier and Zimmermann mentioned above.

Braid groups and quiver mutation

“…Remark 4.19. The arguments of [Gra2] generalize to show that, if vertices i and j of Q are joined by an arrow, then F i F j F i ∼ = F j F i F j can be realized as a single periodic twist. Similarly, one can show that if i → j → k → i is a 3-cycle in Q then F 1 F 2 F 3 F 1 ∼ = F 2 F 3 F 1 F 2 ∼ = F 3 F 1 F 2 F 3 can be realized as a single periodic twist.…”

“…We now describe the braid relations for spherical twists, as in Propositions 2.12 and 2.13 of [ST] (see also [RZ,Gra2]). So we have a group homomorphism ρ : Si ) .…”

Higher zigzag algebras

“…1 = ϕ * 1 : V * ∼ → V * . Note that we do not invert ϕ * 1 , so our formulae will be different to those in [Gra15]. To translate, use the bimodule isomorphisms…”

“…Periodic twists were introduced in [Gra12] as a generalization of the spherical twists for projective modules over symmetric algebras described above. They were later used to study actions of longest elements in braid groups, using a lifting theorem, in [Gra15]. The construction given there is as follows.…”

Diagrammatics for Coxeter groups and their braid groups

“…One expects these to arise from a categorical action of the Hecke algebra, which would then imply the strictness of these actions. Spherical twists such as those in [31] can be investigated using technology developed by Joseph Grant [17,16]. In upcoming work of the first author and Grant, we will demonstrate the connection between certain actions by spherical twist and categorical Hecke algebra actions.…”