Combinatorics of the "coincidental" reflection groups

Much modern combinatorics involves finite reflection groups, both real and complex. Mysteriously, many results work particularly well for the so-called "coincidental" reflection groups. These are the groups generated by n reflections acting in n-dimensional space whose exponents form an arithmetic sequence – they are the real reflection groups of types A, B/C, H3, dihedral groups, and all non-real Shephard groups (the symmetries of regular complex polytopes). This talk will discuss recent work featuring the coincidental groups. This includes recent work with Shepler and Sommers uncovering their extra elegant invariant theory, leading to product formulas for the face numbers and h-vectors of their associated cluster complexes, and a q-analogue of the transformation taking the h-vector to f-vector. We also hope to discuss theorems and conjectures of various others, such as Alex Miller, Barnard–Reading, Hamaker–Patrias–Pechenik–Williams, and Sam Hopkins.