Two Algorithms to Compute Symmetry Groups for Landau-Ginzburg Models
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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 maximal symmetry group G_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. A previously conjectured algorithm to address this problem relies on intersecting groups generated from 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^ via this isomorphism, and show its efficiency in all cases.
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