Discrete torsion, symmetric products and the Hubert scheme

Recently the understanding of the cohomology of the Hubert scheme of points on K3 surfaces has been greatly improved by Lehn and Sorger [18]. Their approach uses the connection of the Hilbert scheme to the orbifolds given by the symmetric products of these surfaces. We introduced a general theory replacing cohomology algebras or more generally Frobenius algebras in a setting of global quotients by finite groups [14]. This is our theory of group Probenius algebras, which are group graded non-commutative algebras whose non-commutativity is controlled by a group action. The action and the grading turn these algebras into modules over the Drinfel’d double of the group ring. The appearance of the Drinfel’d double is natural from the orbifold point of view (see also [17]) and can be translated into the fact that the algebra is a G-graded G-module algebra in the following sense: the G action acts by conjugation on the grading while the algebra structure is compatible with the grading with respect to left multiplication (cf. [16, 20]).