Proof of the $$K(\pi , 1)$$ conjecture for affine Artin groups

We prove the $$K(\pi ,1)$$ K ( π , 1 ) conjecture for affine Artin groups: the complexified complement of an affine reflection arrangement is a classifying space. This is a long-standing problem, due to Arnol’d, Pham, and Thom. Our proof is based on recent advancements in the theory of dual Coxeter and Artin groups, as well as on several new results and constructions. In particular: we show that all affine noncrossing partition posets are EL-shellable; we use these posets to construct finite classifying spaces for dual affine Artin groups; we introduce new CW models for the orbit configuration spaces associated with arbitrary Coxeter groups; we construct finite classifying spaces for the braided crystallographic groups introduced by McCammond and Sulway.

On the -problem for restrictions of complex reflection arrangements

Let $W\subset \operatorname{GL}(V)$ be a complex reflection group and $\mathscr{A}(W)$ the set of the mirrors of the complex reflections in $W$. It is known that the complement $X(\mathscr{A}(W))$ of the reflection arrangement $\mathscr{A}(W)$ is a $K(\unicode[STIX]{x1D70B},1)$ space. For $Y$ an intersection of hyperplanes in $\mathscr{A}(W)$, let $X(\mathscr{A}(W)^{Y})$ be the complement in $Y$ of the hyperplanes in $\mathscr{A}(W)$ not containing $Y$. We hope that $X(\mathscr{A}(W)^{Y})$ is always a $K(\unicode[STIX]{x1D70B},1)$. We prove it in case of the monomial groups $W=G(r,p,\ell )$. Using known results, we then show that there remain only three irreducible complex reflection groups, leading to just eight such induced arrangements for which this $K(\unicode[STIX]{x1D70B},1)$ property remains to be proved.

The Jacobian, Reflection Arrangement and Discriminant for Reflection Hopf Algebras

We study finite-dimensional semisimple Hopf algebra actions on noetherian connected graded Artin–Schelter regular algebras and introduce definitions of the Jacobian, the reflection arrangement, and the discriminant in a noncommutative setting.

Hyperfactored of Reflection Arrangement A�(G_25 )

The purpose of this paper is to study the hyperfactored of the complex reflection arrangementA(G 25 ). Depending on the lattice of arrangement A(G 25 ), the basis of A(G 25 ) has been foundand then partitioned. Also, showed that A(G 25 ) is not hyperfactored and is not inductivelyfactored.

Three Counter-Examples on Semi-Graphoids

Semi-graphoids are combinatorial structures that arise in statistical learning theory. They are equivalent to convex rank tests and to polyhedral fans that coarsen the reflection arrangement of the symmetric group Sn. In this paper we resolve two problems on semi-graphoids posed in Studený's book (2005), and we answer a related question of Postnikov, Reiner and Williams on generalized permutohedra. We also study the semigroup and the toric ideal associated with semi-graphoids.