Two Algorithms to Compute Symmetry Groups for Landau-Ginzburg Models

Landau-Ginzburg mirror symmetry studies isomorphisms between graded Frobenius algebras, known as A- and B-models. Fundamental to constructing these models is the computation of the finite, Abelian $\textit{maximal symmetry group}$ $G_{W}^{\max}$ of a given polynomial $W$. For $\textit{invertible}$ polynomials, which have the same number of monomials as variables, a generating set for this group can be computed efficiently by inverting the $\textit{polynomial exponent matrix}$. However, this method does not work for $\textit{noninvertible}$ polynomials with more monomials than variables since the resulting exponent matrix is no longer square. A previously conjectured algorithm to address this problem relies on intersecting groups generated from $\textit{submatrices}$ of the exponent matrix. We prove that this method is correct, but intractable in general. We overcome intractability by presenting a group isomorphism based on the Smith normal form of the exponent matrix. We demonstrate an algorithm to compute $G_{W}^{\max}$ via this isomorphism, and show its efficiency in all cases.