  We take new algebraic and geometric perspectives on the combinatorial results
recently obtained on the partition functions of critical massive gravities
conjectured to be dual to Logarithmic CFTs throught the AdS$_3$/LCFT$_2$
correspondence. We show that the partition functions of logarithmic states can
be expressed in terms of Schur polynomials. Subsequently, we show that the
moduli space of the logarithmic states is the symmetric product $S^n \left(
\mathbb{C}^2 \right)$. As the quotient of an affine space by the symmetric
group, this orbifold space is shown to be described by Hilbert series that have
palindromic numerators. The palindromic properties of the Hilbert series
indicate that the orbifolds are Calabi-Yau, and allow for a new interpretation
of the logarithmic state spaces in critical massive gravities as Calabi-Yau
singular spaces.
