Lifts of longest elements to braid groups acting on derived categories

TL;DR

This paper investigates how lifts of the longest elements in symmetric groups act on derived categories via braid group actions, providing a general framework using periodic twists and extending previous special case results.

Contribution

It introduces a lifting theorem for periodic twists, enabling the extension of known actions to more general derived categories of symmetric algebras.

Findings

Established a lifting theorem for periodic twists

Provided new proofs for braid relation existence

Extended previous results to general symmetric algebra cases

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.