Linear quotient families and stabilizer posets

We are concerned with the stabilizer poset of linear actions of finite groups. This is originally motivated through our attempt to describe the explicit geometry of the universal families over moduli spaces of Riemann surfaces. Here these universal families are locally approximated by linear quotient families associated with the linear actions of the automorphism groups of Riemann surfaces on the vector spaces of holomorphic quadratic differentials. To describe such families, the stabilizers for these linear actions play an important role. For instance, in these families, the fibers over stabilizer-constant loci are identical (the quotient fiber theorem). We in fact study the stabilizer posets, because they correspond to the posets of stabilizer-constant loci under the geometric Galois correspondence. We provide an algorithm to determine these stabilizer posets—in fact it works for the stabilizer posets for any linear action of any finite group. This algorithm is based on linear algebra combined with maximal conjugacy classes of stabilizers and is quite powerful in practical computation.