Simple Symmetric Matrix Singularities and the Subgroups of Weyl Groups Aμ, Dμ, Eμ

We analyse the classification of simple symmetric matrix singularities depending on two parameters which was obtained recently by Bruce and Tari [2]. We show that these singularities are classified by certain reflection subgroups Y of the Weyl groups X = Aμ, Dμ, Eμ. The Dynkin diagram of such a subgroup is obtained from the affine diagram of X by deleting vertices of total marking 2: deletion of two 1-vertices corresponds to a 2× 2 matrix singularity, and deletion of one 2-vertex gives rise to a 3× 3 matrix. The correspondence is based on an isomorphism of the discriminants and on the description of a relevant monodromy group of the determinantal curve. Moreover, the base of a miniversal deformation of a simple matrix singularity turns out to be isomorphic to the quotient of the complex configuration space of the group X by the subgroup Y . We discuss lattice properties of symmetric matrix families in two variables which, in the case of simple singularities, define the choice of the subgroups. 2000 Math. Subj. Class. 58C27, 53A25, 65F15, 58K.