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I am currently reading the book Spin Geometry by Lawson/Michelsohn to understand Dirac Operators and related topics. At some point it uses representation theory to classify Clifford Algebras. In particular this book states in I,§5. Theorem 5.6 on p.31:

Let $K=\mathbb{R}, \mathbb{C}$ or $\mathbb{H}$ and consider the ring $K(n)$ of $n \times n$ $K$-matrices as an algebra over $\mathbb{R}$. Then the natural representation $\rho$ of $K(n)$ on the vector space $K^n$ is, up to equivalence, the only irreducible representation of $K(n)$. The algebra $K(n) \oplus K(n)$ has exactly two equivalence classes of irreducible representations. They are given by $$ \rho_1(\varphi_1,\varphi_2) = \rho(\varphi_1) \text{ and } \rho_2(\varphi_1,\varphi_2) =\rho(\varphi_2) $$ acting on $K(n)$.

I am perfectly fine with the statement, but I would like to see a proof of that. Lawson/Michelsohn claim that this follows from the fact that the algebras $K(n)$ are simple and that simple algebras have only one irreducible representation up to equivalence. For the details the reader is referred to Lang: Algebra. Since this book as nearly 1000 pages I would like to ask, if someone could be a little more precise? Of course I would also be happy with a direct proof of the claim or a reference to any other readably textbook containing a proof. I must point out that I have no prior experience in the field of representation theory.

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Wait a sec, it's not clear from your question: do you want a reference to something other than Lang? Even if so, I encourage you to read just the section of Lang containing the theorems mentioned below; you will not find a more self-contained treatment. –  Stephen Mar 7 '12 at 1:50
As I said, I would be happy with either a direct proof, a more precise reference to Lang or a precise reference to another textbook. –  Meneldur Mar 8 '12 at 16:21

2 Answers 2

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The specific reference: look at Theorems 4.3 and 4.4 on page 653 of the revised third edition (does that make it edition 3.5?) of Lang's "Algebra". These give you precisely the statements you want. The proofs are pretty readable, in my opinion, but if you like I will give more details here.

To apply these theorems to your situation, observe that each of the matrix algebras you are looking at is the direct sum of $n$ left ideals, the $i$th of which consists of matrices which have non-zero entries only in the $i$th column. Each of these ideals is simple, and they are all isomorphic as left modules to $K^n$. This is the definition of simple ring given by Lang: it is a semisimple ring (that is, is a direct sum of simple ideals) with only one isoclass of left ideals appearing in the direct sum decomposition. This also shows that the direct sum of two copies of your matrix algebra is semisimple, and Lang's Thm 4.4 gives the statement about its representations that you need.

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Thanks, now I see much better, why Lang is helpful here. –  Meneldur Mar 8 '12 at 16:22

In the case $\mathbb{K} = \mathbb{C}$, I believe there are in fact two distinct equivalence classes of (complex) irreducible representations, namely the defining rep and its complex conjugate ($ M \in \mathbb{C}(n)$ acting on $\mathbb{C}^n$ by multiplying by $\overline{M}$ on the left). Whilst these are inequivalent as representations, they do however result in isomorphic $\mathbb{C}(n)$-modules, consistent with Lang (corollary 4.6 on page 653).

The distinction is that an equivalence of complex reps requires that the predicted $\mathbb{C}$-module isomorphism be complex-linear, though it's pretty straightforward to prove that it must in fact be antilinear.

To show that $\mathbb{R}(n)$ has a unique real irreducible rep is easy and I believe the quaternionic case is unique as well, though I haven't seen a proof.

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