Nichols-Woronowicz model of coinvariant algebra of complex reflection groups

Let V be a finite dimensional complex vector space. A finite subgroup G ⊂ GL(V ) is called a complex reflection group, if G can be generated by the set of pseudoreflections, i.e., transformations that fix a complex hyperplane in V pointwise. Any real reflection group becomes a complex reflection group if one extends the scalars from R to C. In particular all Coxeter groups give examples of complex reflection groups. We refer the reader to [3] for general background of the theory of complex reflection groups. Below we recall a few facts about real and complex reflection groups which appeared to be a motivation for our paper. In 1954, G. C. Shephard and J. A. Todd [16] had obtained a complete classification of finite irreducible complex reflection groups. They found that there exist an infinite family of irreducible complex reflection groups G(e, p, n) depending on three positive integer parameters (with p dividing e), and 34 exceptional groups G4, . . . , G37. The group G(e, p, n) has the order enn!/p. It also has a normal abelian subgroup of order en/p, and the corresponding quotient is the symmetric group on n points. The family of groups G(e, p, n) includes the cyclic group Ce/p of order e/p, namely, Ce/p = G(e, p, 1); the symmetric group on n points Sn = G(1, 1, n); the Weyl groups of types Bn, Cn, and Dn, namely, WBn = WCn = G(2, 1, n) and WDn = G(2, 2, n); and the dihedral groups I2(e) = G(e, e, 2). The fundamental fact characterizing the finite complex reflection subgroups in GL(V ) is the following theorem by G. C. Shephard and J. A. Todd.