Here's the question in a nutshell. For some $n\in\mathbb N$, we consider the polynomial

$\det\left(\left(X_{i,j}\right) _ {1\leq i\leq n,\ 1\leq j\leq n}\right)\in\mathbb Z\left[X_{i,j}\mid 1\leq i\leq n,\ 1\leq j\leq n\right]$

in $n^2$ indeterminates $X_{i,j}$. This is known to be irreducible over $\mathbb Z$, but is there a "nice" ring in which $\mathbb Z$ embeds and where this polynomial splits into linear factors? The ring needs not be commutative, but the variables $X_{i,j}$ are still supposed to commute with everything from this ring. For instance, if $n=1$, then the ring can be taken to be $\mathbb Z$, and if $n=2$, then it can be taken to be $M_2\left(\mathbb Z\right)$, since

$\det\left(\begin{array}{cc}X&Y \newline Z&W\end{array}\right) = \left(\begin{array}{cc}X&Y \newline Z&W\end{array}\right)\mathrm{adj}\left(\begin{array}{cc}X&Y \newline Z&W\end{array}\right)$,

and both factors on the right hand side are linear in $X$, $Y$, $Z$, $W$. For $n>2$, however, the adjoint is not linear anymore. Can we extend $M_n\left(\mathbb Z\right)$ further to make it split? (This is what I mean by "nice" - it should be a kind of natural generalization of $M_n\left(\mathbb Z\right)$. Although I have troubles constructing even a non-nice splitting ring...)

Here is the actual source of the question:

Keith Conrad, in his expository paper *The Origin of Representation Theory*, discusses an apparently forgotten problem that goes back to Dedekind: Given a finite group $G$, the polynomial $\det\left(\left(X_{gh^{-1}}\right)_{g\in G,\ h\in G}\right)\in\mathbb Z\left[X_g\mid g\in G\right]$ (this is a generalization of the circulant, which is obtained if $G$ is a cyclic group) is known to split into a product of irreducible factors as follows:

$\det\left(\left(X_{gh^{-1}}\right) _ {g\in G,\ h\in G}\right) = \prod\limits_{\rho\text{ is an irrep of }G\text{ over }\mathbb C} \det\left(\sum\limits_{g\in G}X_g\rho\left(g\right)\right)^{\dim\rho}$.

(Okay, apparently Keith writes $\deg \rho$ instead of $\dim \rho$, but otherwise this is in his Section 5.)

Now, some of the factors on the right hand side - those corresponding to representations of dimension $> 1$ - are nonlinear, and Dedekind tried to split them into linear factors by extending the base field. Two examples are given, and both times the extension of the field is more or less the endomorphism ring of the representation $\rho$ - but this is not surprising, because both times $\dim \rho=2$, and we have the adjoint decomposition I gave above for the case $n=2$. The actually interesting problem seems to be the $n > 2$ case. Since any irrep $\rho$ over an algebraically closed field like $\mathbb C$ has the property that the $\rho\left(g\right)$ for all $g\in G$ span the whole endomorphism ring of the irrep (this is called the density theorem, I believe), we can actually forget about the irrep and try to split the determinant of the general matrix. That's the problem above.