  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.
