For every real reflection group, one can define several polynomials from both combinatorial and geometric data that exhibit the same remarkable factorization over the integers, involving the exponents of the group. This paper surveys in separate sections various aspects connected to a possible combinatorial interpretation of such factorizations, connecting them to the topology of the associated Lie group, the geometry of real and complex reflection groups, the (combinatorially defined) length functions on the groups, the characteristic polynomial of the associated geometric lattice, the broken circuit complex of the associated matroid, and the generalized exponents in Terao’s theory of “free” arrangements. The final section reports about recent attempts to obtain the exponents of Coxeter groups from entirely combinatorial methods.
For the entire collection see [Zbl 0806.00023].