# splitting spin representation when reducible

I'm looking at the spin representation of the orthogonal algebra $so(2m,C)$. This representation is $2^m$ dimensional and an explicit construction with gamma matrices is well known (see for example http://hitoshi.berkeley.edu/230A/clifford.pdf). It is also well known that this is reducible into two $2^{m/2}$ dimensional irredible reps. I'd like to get an explicit construction of these two irreps as matrices... similar to the construction in the link above. Does anyone know where that's give.

• First express the $\gamma_i$ matrices into a chiral representation, that is with the left-up and right-down blocks being zero. Then build the $so(2m)$ generators $\omega_{ij} =[\gamma_i, \gamma_j]$ which will be block-diagonal. – Trimok Jul 16 '13 at 9:30

Your Link already shows how to build up the matrix representations by building the tensor product as indicated (Equations (3) to (12) in Hitoshi script). For example in the popular four dimension $N=2k=2\cdot 2$, so $k=2$ and on build the matrix representation out of 2 Pauli matrices. The dimension of the constituents of this tensor product are 2 by 2 matrices, so their product will result in 4 by 4 matrices:

$\gamma^{1}=\sigma_{1}\otimes\mathbb{1}=\left(\begin{matrix}0&1\\1&0\end{matrix}\right)\otimes\left(\begin{matrix}1&0\\0&1\end{matrix}\right)=\left(\begin{matrix}0\cdot\left(\begin{matrix}1&0\\0&1\end{matrix}\right)&1\cdot\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\\1\cdot\left(\begin{matrix}1&0\\0&1\end{matrix}\right)&0\cdot\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\end{matrix}\right)=\left(\begin{matrix}0&0&1&0\\0&0&0&1\\1&0&0&0\\0&1&0&0\end{matrix}\right)$,

$\gamma^{2}=\sigma_{2}\otimes\mathbb{1}=\left(\begin{matrix}0&-i\\i&0\end{matrix}\right)\otimes\left(\begin{matrix}1&0\\0&1\end{matrix}\right)=\left(\begin{matrix}0\cdot\left(\begin{matrix}1&0\\0&1\end{matrix}\right)&\left(-i\right)\cdot\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\\i\cdot\left(\begin{matrix}1&0\\0&1\end{matrix}\right)&0\cdot\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\end{matrix}\right)=\left(\begin{matrix}0&0&-i&0\\0&0&0&-i\\i&0&0&0\\0&i&0&0\end{matrix}\right)$,

$\gamma^{3}=\sigma_{3}\otimes\sigma_{1}=\left(\begin{matrix}1&0\\0&-1\end{matrix}\right)\otimes\left(\begin{matrix}0&1\\1&0\end{matrix}\right)=\left(\begin{matrix}1\cdot\left(\begin{matrix}0&1\\1&0\end{matrix}\right)&&0\cdot\left(\begin{matrix}0&1\\1&0\end{matrix}\right)\\0\cdot\left(\begin{matrix}0&1\\1&0\end{matrix}\right)&&-1\cdot\left(\begin{matrix}0&1\\1&0\end{matrix}\right)\end{matrix}\right)=\left(\begin{matrix}0&1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&-1&0\end{matrix}\right)$,

$\gamma^{4}=\sigma_{3}\otimes\sigma_{2}=\left(\begin{matrix}1&0\\0&-1\end{matrix}\right)\otimes\left(\begin{matrix}0&-i\\i&0\end{matrix}\right)=\left(\begin{matrix}1\cdot\left(\begin{matrix}0&-i\\0&i\end{matrix}\right)&0\cdot\left(\begin{matrix}0&-i\\i&0\end{matrix}\right)\\0\cdot\left(\begin{matrix}0&-i\\i&0\end{matrix}\right)&\left(-1\right)\cdot\left(\begin{matrix}0&-i\\i&0\end{matrix}\right)\end{matrix}\right)=\left(\begin{matrix}0&-i&0&0\\i&0&0&0\\0&0&0&i\\0&0&-i&0\end{matrix}\right)$,

$\gamma^{5}=\left(-i\right)^{2}\gamma^{1}\gamma^{2}\gamma^{3}\gamma^{4}$.

A particular good summary about spinors and gamma matrices (Spinor representations of $SO(n,m)$) in various dimensions can be found in Polchinski´s String Theory part II, Appendix B. There you also find in which dimensions $2^{\frac{m}{2}}\times 2^{\frac{m}{2}}$ Weyl representations of the gamma matrices are even possible in the first place.

Alternatively I could suggest you to look into the book Gravity and Strings by Thomas Ortin in Appendix B which provides explicit representations for irreducible spinor representations in various dimensions.

• This is the construction of the full spin representation. I have no trouble following the construction in the link but this rep is reducible so there should be two irreps each consisting of 2x2 matrices...it's these 2x2 matrices that I'm after. Similarly for so(8,C) the full spin rep as constructed in the link is 16-dimensional. I'm looking for the two 8 dim irreps that this splits to...so two sets of 8x8 matrices.. – Y M Jul 16 '13 at 0:49