Take a ring $Z_n = \{0,1,...,n-1\}$. All operations occur modulo $n$. Consider the $n^{th}$ roots of unity $\omega_n = \exp\left(\frac{2\pi i}{n}\right)$. Now define
$$X = \sum_a \vert{a+1}\rangle\langle{a}\vert$$ $$Z = \sum_a \omega_a \vert{a}\rangle\langle{a}\vert,$$
where $\vert a\rangle$ is a column vector of an orthonormal basis and $\langle a\vert$ is the corresponding basis row vector.
These matrices follow the commutation $XZ = \omega ZX$. The Weyl matrices are defined as $W_{a,b} = X^aZ^b$.
Vectorization of a matrix $A=\sum A_{i,j}\vert i \rangle\langle j\vert$ is defined by vec$(A) = \sum A_{i,j}\vert i \rangle\vert j\rangle$
I am given an identity vec$(\mathbb{I})$vec$(\mathbb{I})^* = \frac{1}{n}\sum\limits_{c,d}\overline{W_{c,d}}\otimes W_{c,d}$. Could someone point out how I could prove this?