I'm wondering if anybody knows where one can find an English translation of Emmy Noether's classical paper E. NOETHER, Hyperkomplexe Grössen und Darstellungstheorie, Math. Zeit. 30(1929), 641–692 ?



English translation of Emmy Noether's Hyperkomplexe Grössen und Darstellungstheorie
E. NOETHER, Hyperkomplexe Grössen und Darstellungstheorie [Hypercomplex Quantities and the Theory of Representations], Math. Zeit. 30 (1929), 641–692; informal translation of section 23.



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[original](https://ilorentz.org/beenakker/MO/Noether_par_23.pdf)  

**Determinant of a hypercomplex system.**


Let $\mathbb{o}=a\_1P+\ldots+a\_h P$ be a hypercomplex system. I adjoin to $P$ a number $h$ of undetermined $x\_1,\ldots x\_h$ and construct  

[equation]  

The $x$ should be interchangeable with the $a\_i$; therefore the computational rules of $\mathbb{o}^\ast$ are already determined.


In $\mathbb{o}^\ast$ there is a "general element of $\mathbb{o}$"  

[equation]  

If in a certain representation $a\_i\mapsto A\_i$, then $w$ is associated to  

[equation]  

$W$ is called the *system matrix* of the representation (or in particular, when the $a\_i$ build a group with ring ${\mathbb{o}}$, the *group matrix*). If the representation is regular, then one has a *regular system matrix*.  

The elements $w\_{ik}$ of $W$ are linear forms of $x$. The "*system determinant*" $|W|$ is therefore of degree $n$, when the representation has degree $n$. In particular, the *regular system determinant* is of degree $h$.  

The system determinant does not change when one changes to an equivalent representation, since  

[equation]  

In the transition from $(a\_1,\ldots,a\_h)$ to a new basis $(b\_1,\ldots b\_h)$ and from $w=\sum a\_ix\_i$ to $w=\sum b\_iy\_i$ one finds new elements of $W$ and therefore also a new determinant, when in the old determinant one substitutes  

[equation]  

with a regular substitution matrix.


If one has a decomposition of the representation module, the matrix $W$ in a suitable coordinate system would look like this  

[equation]  

The determinants $|W\_i|$ of a particular representation are the same as in the regular representation of $\mathbb{o}$ or even of $\mathbb{o}/\mathbb{c}$, with $\mathbb{c}$ a radical.


*If $P$ is algebraically closed, then the determinant $|W\_i|$ belonging to an irreducible representation is a prime function in $x$, and inequivalent representations belong to distinct prime factors.*


**Proof:** Because all irreducible representations of $\mathbb{o}$ are also representations of $\mathbb{o}/\mathbb{c}$, we can restrict ourselves to the ring without the radical $\mathbb{c}$. In this ring we take as basis the matrix unities $c\_{ik}$; the general element then takes the form:  

[equation]  

The matrices of the irreducible representations are: $W\_\nu=(x\_{ik}^{(\nu)})$. The functions $|W\_\nu|=|x\_{ik}^{(\nu)}|$ are known to be irreducible and evidently distinct.  

To calculate the $|W\_\nu|$, one can decompose the regular system matrix of $\mathbb{o}/\mathbb{c}$ into prime factors. Each prime factor appears as often as the degree of the irreducible representation, because the corresponding ideal $\mathbb{l}\_\nu$ appears equally often in the decomposition of the representation module.  

One could also start from the regular system matrix of $\mathbb{o}$, but then one would obtain each irreducible factor more often, namely as often as the corresponding $\mathbb{l}\_\nu$ appears as composition factor in the left-ideal. With this regular representation we considered $\mathbb{o}$ as the left-ideal; if instead one would consider $\mathbb{o}$ as the right-ideal, then one obtains a second regular system matrix (the "antistrophe matrix" of Frobenius), which has the same irreducible factors (namely, the system determinants of the collected irreducible representations). Possibly, the exponents of the irreducible factors are different (see the example in paragraph 10).  

The system determinants of a commutative system decompose into linear factors, because all irreducible representations are of degree one. These linear factors are themselves the irreducible representations, so they are the characters of an Abelian group. This fact was the starting point of Dedekind's study of the group determinants of non-Abelian groups.


