Categorical braid group actions and cactus groups
Iva Halacheva, Anthony Licata, Ivan Losev, and Oded Yacobi

TL;DR
This paper explores the action of braid and cactus groups on categorical representations of quantum groups, establishing t-exactness and perverse equivalences, and connecting these actions to crystal bases and symmetric group representations.
Contribution
It proves t-exactness of Rickard complexes for isotypic representations, constructs cactus group actions on crystals, and links categorical actions to classical symmetric group actions on Kazhdan-Lusztig bases.
Findings
Proves t-exactness of braid group actions for isotypic cases.
Constructs cactus group actions on crystal bases.
Provides new proofs of symmetric group actions on Kazhdan-Lusztig bases.
Abstract
Let be a semisimple simply-laced Lie algebra of finite type. Let be an abelian categorical representation of the quantum group categorifying an integrable representation . The Artin braid group of acts on by Rickard complexes, providing a triangulated equivalence , where is a weight of and is a positive lift of the longest element of the Weyl group. We prove that this equivalence is t-exact up to shift when is isotypic, generalising a fundamental result of Chuang and Rouquier in the case . For general , we prove that is a perverse equivalence with respect to a Jordan-H\"older filtration of . Using these results we construct, from the action…
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Algebra and Geometry · Molecular spectroscopy and chirality
