For $\mu=0,1,2,3$, let $\gamma^{\mu}$ the set of Dirac gamma matrices. What does it mean to say that $\{\gamma^{\mu}: \hspace{0.1cm} \mu=0,1,2,3\}$ is irreducible?


From my [previous question](https://mathoverflow.net/questions/386335/what-is-the-relationship-between-the-dirac-algebra-and-the-clifford-algebra), I know that the gamma matrices determine a representation $\rho$ of the Clifford algebra $\mathcal{C}l\_{1,3}(\mathbb{R}^{4},\Phi)$ in the algebra of $4\times 4$ complex matrices. Many references state that the gamma matrices are irreducible representations of $\mathcal{C}l\_{1,3}(\mathbb{R}^{4},\Phi)$, which means that for every $x \in \mathcal{C}l\_{1,3}(\mathbb{R}^{4},\Phi)$, $\rho(x)$ has no non-trivial invariant subspaces.


However, as discussed [here](https://physics.stackexchange.com/questions/138335/dirac-group-representation), the gamma matrices can also determine a group $\Gamma\_{\text{Dirac}}$ with elements $I, \pm \gamma^{\mu}, \pm \gamma^{\mu}\gamma^{\nu}$, $\mu = 0,1,2,3,5$ and $\mu < \nu$. Here $\gamma^{5} := \gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$. And the notion of irreducible representation also holds for groups.


So, getting back from the original question, when one says $\{\gamma^{\mu}: \hspace{0.1cm} \mu=0,1,2,3\}$, what does it mean exactly? Maybe both notions defined above have some natural connection?


**ADD:** *I think* I got it. It can be shown that the representation $\rho : \mathcal{C}l\_{1,3}(\mathbb{R}^{4},\Phi) \to M\_{4\times 4}(\mathbb{C})$ is not only faithful, but also a bijection. Thus, because it is surjective, the representation is irreducible. Now, each element of the Dirac group $\Gamma\_{\text{Dirac}}$ is, in particular, the image $\rho(x)$ of an element $x \in \mathcal{C}l\_{1,3}(\mathbb{R}^{4},\Phi)$ under the representation $\rho$. Thus, there is no (non-trivial) subspace of $\mathbb{C}^{4}$ which is invariant under the action of any of these elements. Hence, the representation $\tilde{\rho}: \Gamma\_{\text{Dirac}} \to M\_{4\times 4}(\mathbb{C})$ given by $\tilde{\rho}(x) = x$ must be irreducible. In summary, because $\rho$ is irreducible, the group representation $\tilde{\rho}$ is irreducible.


Any comments and corrections are welcome.


**EDIT:** I actually don't know if the surjectivity of $\rho$ implies $\rho$ to be irreducible.


**ADD 2:** To clarify, the space $M\_{4\times 4}(\mathbb{C})$ mentioned before is being considered as a vector space over $\mathbb{R}$, as well as $\mathcal{C}l\_{1,3}(\mathbb{R}^{4},\Phi)$. Surely one can complexify $\mathcal{C}l\_{1,3}(\mathbb{R}^{4},\Phi)$ to get a new Clifford algebra $\mathcal{C}l\_{1,3}(\mathbb{R}^{4},\Phi)\_{\mathbb{C}}$ which now is isomorphic to $M\_{4\times 4}(\mathbb{C})$, the latter being considered as an algebra over $\mathbb{C}$.


After posting this question, some ideas and questions have arisen. Let me summarize the important points/questions below:


* It is not clear to me yet if the irreducibility of $\rho: \mathcal{C}l\_{1,3}(\mathbb{R}^{4},\Phi)\to M\_{4\times 4}(\mathbb{C})$ is equivalent to the irreducibility of $\tilde{\rho}: \Gamma\_{\text{Dirac}} \to M\_{4\times 4}(\mathbb{C})$ in any way. However, as I tried to prove above, the irreducibility of $\rho$ seems to imply the irreducibility of $\tilde{\rho}$.
* It is also not clear if the proof I give before is correct, that is, I don't know if the bijectivity of $\rho$ is enough to garantee that it is irreducible.
* In the physics literature, it is sometimes not clear if the matrix algebra is being considered as a *real* or *complex* algebra, so I don't if the representation from $\rho: \mathcal{C}l\_{1,3}(\mathbb{R}^{4},\Phi) \to M\_{4\times 4}(\mathbb{C})$ is irreducible *or* if we need to complexify $\mathcal{C}l\_{1,3}(\mathbb{R}^{4},\Phi)$ to get a irreducible representation $\hat{\rho}: \mathcal{C}l\_{1,3}(\mathbb{R}^{4},\Phi)\_{\mathbb{C}}\to M\_{4\times 4}(\mathbb{C})$. Some references suggest that the complex case is irreducible, and I wasn't able to find a proof of this fact or some indiciation if the real case is also irreducible.



Gamma matrices are irreducible
The irreducible representations of the Clifford algebra $C$ and the associated group $\Gamma$ (Dirac group) are worked out in [these notes.](https://arxiv.org/abs/hep-th/9811101)


In *even* $D$ space-time dimensions (in particular, for $D=3+1=4$) the group $\Gamma$ has $2^D+1$ inequivalent irreducible representations. Of these $2^D$ are one-dimensional, and the remaining representation has dimension $2^{D/2}$. The Clifford algebra itself has only one irreducible representation and this has dimension $2^{D/2}$. This spinor representation is constructed in the cited notes (section 1.2). For $D=4$ with the signature $(-1,+1,+1,+1)$ these are the four Dirac matrices.


