Abstract: Combinatorialists love to turn their favorite objects (graphs, posets, permutations) into hyperplane arrangements and their regions. The Braid, Shi, and Catalan arrangements, and their analogues for Weyl groups, have been particularly popular examples with remarkable numerological and structural properties: their regions can be labeled by trees, their characteristic polynomials factor with positive integer roots, they can be extended to free filtrations of the affine Weyl arrangements.
Representation theorists and geometers love them all the same: These three types of arrangements are used to study Kazhdan-Lusztig cells and GIT stability conditions, and the polynomial vector fields tangent to them form free modules with a very rigid structure.
We will introduce a new large family of generalizations of these arrangements, with surprisingly good behavior, that we constructed in recent work studying restrictions on arbitrary flats. We will present numerological and structural results for them and relate them to the representation theory of parking spaces and rational Cherednik algebras, and to ramification formulas associated to the braid monodromy of the W-discriminant hypersurfaces. If time permits, we will also discuss a connection to periodic and aperiodic tilings and work joint with Olivier Bernardi, where we give refined combinatorial models for their regions.