2025/10/09 by Davide Dal Martello, Martello, Davide Dal
Mathematics · #20C08 (Secondary) #20F05 #20F36 #20F55 (Primary) #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Group Theory (math.GR) #Quantum Algebra (math.QA) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2510.08209
openalex publication_date 2025/10/09 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28
Aiming for a revival of the theory of crystallographic complex reflection groups, we compute (minimal) Coxeter-like reflection presentations for the infinite families of those non-genuine groups which satisfy Steinberg's fixed point theorem. These new presentations behave à la Coxeter, encoding many of the group's properties at a glance, and their signature feature -- named the x-relation -- is fully understood in terms of configuration spaces. Crucially, the presentations further achieve the braid theorem, allowing to deform into the generic Hecke algebra. In particular, we revisit the affine GDAHA family in deformation terms of the most general class of Steinberg crystallographic complex reflection groups.