Two Algorithms to Compute the Maximal Symmetry Group

Landau-Ginzburg mirror symmetry studies isomorphisms between graded Frobenius algebras, known as Aand B-models. Fundamental to constructing these models is the computation of the finite, Abelian maximal symmetry group Gmax W of a given polynomial W . For invertible polynomials, which have the same number of monomials as variables, a generating set for this group can be computed efficiently by inverting the polynomial exponent matrix. However, this method does not work for noninvertible polynomials with more monomials than variables since the resulting exponent matrix is no longer square. In this paper we present and analyze two characterizations of the maximal symmetry group that address this problem—one based on submatrices of the exponent matrix, and the other based on the Smith normal form of the exponent matrix. We analyze the resulting algorithms based on these characterizations, demonstrating the efficiency of the latter and the intractability of the former.