$V_4$ can be used to give a homological characterisation of the parity of the dimensions of the representations. Proofs of the propositions are given in the last section.

\[prop:heller\] [@ring] If $M$ is an even dimensional indecomposable projective-free representation of $V_4$, then $\Omega(M) \cong M$.

A finite dimensional projective-free representation $M$ of $V_4$ is even dimensional if and only if $\Omega(M) \cong M$.

We only have to show that if $M$ is an odd dimensional indecomposable then $\Omega(M) \ncong M$. By the classification theorem, we know that $M$ is isomorphic to $\Omega^l(k)$ for some integer $l$. Then $\Omega(M) \cong \Omega(\Omega^l(k)) \cong
\Omega^{l+1}(k)$, which is clearly not isomorphic to $M$ just for dimensional reasons: just note that dimension of $\Omega^{n} (k)$ is $2n+1$.

Dual representations of $V_4$
-----------------------------

We will use our knowledge of the representations of $V_4$ to characterise the parity of the dimension of a representation using the concept of self-duality.

Recall that if $M$ is a finite dimensional representation of a group $G$, then one can talk about the dual representation $M^* := \operatorname{Hom}(M,
k)$, where a group element $g$ acts on a linear functional $\phi$ by $(g \cdot \phi)(x) := \phi(x g^{-1})$. A representation of $G$ is 