Transposing Noninvertible Polynomials

Abstract

Landau-Ginzburg mirror symmetry predicts isomorphisms between graded Frobenius algebras (denoted $\mathcal{A}$ and $\mathcal{B}$) that are constructed from a nondegenerate quasihomogeneous polynomial $W$ and a related group of symmetries $G$. Duality between $\mathcal{A}$ and $\mathcal{B}$ models has been conjectured for particular choices of $W$ and $G$. These conjectures have been proven in many instances where $W$ is restricted to having the same number of monomials as variables (called $\textit{invertible}$). Some conjectures have been made regarding isomorphisms between $\mathcal{A}$ and $\mathcal{B}$ models when $W$ is allowed to have more monomials than variables. In this paper we show these conjectures are false; that is, the conjectured isomorphisms do not exist. Insight into this problem will not only generate new results for Landau-Ginzburg mirror symmetry, but will also be interesting from a purely algebraic standpoint as a result about groups acting on graded algebras.