Reflection arrangements and ribbon representations

Real reflection groups are ubiquitous in combinatorics, appearing for example, as symmetries of regular polytopes in real Euclidean space. G.C. Shephard studied an analogous class of complex reflection groups, now known as Shephard groups, which are the symmetries of objects called "regular complex polytopes" inside unitary spaces.

This talk will review these objects, and discuss a generalization of a recent result of Ehrenborg and Jung (that unified previous work of Stanley, Calderbank-Hanlon-Robinson, and Wachs). Ehrenborg and Jung showed why the ribbon/descent representations of Solomon for the symmetric group appear naturally as homology representations within certain subposets of the lattice of set partitions.

Our generalization not only elucidates this geometrically, but also works for all real reflection groups and all complex Shephard groups. It explains why Solomon's ribbon/descent representations and their analogues appear naturally as homology representations within certain subposets of the intersection lattice of their reflection hyperplane arrangements.