New models for arrangements and Artin groups

Mardi, 23. avril 2019 - 14:00 - 15:00

Michael Falk (Arizona)


Given a finite set ${\mathcal A}$ of linear hyperplanes in $\mathbb{R}^n$, we define a partial ordering on the set of pairs of chambers whose nerve has the homotopy type of the complement $M$ of the associated arrangement of complex hyperplanes. This model carries a combinatorial analogue of a circle action that mirrors the action of $\mathbb{C}^*$ on $M$, capturing the complex structure in this special case. If $\mathcal A$ is the set of reflecting hyperplanes of a reflection group $W$, one obtains a model for the quotient $W \backslash M$ as the nerve of the small category generated by the left and right weak orders on $W$. We discuss generalizations and applications to Artin groups and Shephard groups. This is joint work with Emanuele Delucchi, Dana Ernst, and Sonja Riedel.