# Use Cartan subalgebra in spinor representation to find weights of vector representation

For $SO(2n)$ we can construct the lie algebra elements by using antisymmetric combinations of $\gamma_\mu$ which obey the Clifford algebra.

Up to some prefactor the elements $S_{\mu \nu} = \alpha [\gamma_\mu , \gamma_\nu]$ can be used as generators. Then we can identify the cartan subalgebra with the elements $H_i = S_{(2i-1)(2i)}$.

Now I would like to use this to find the weights for an element ($A_\mu)$ in the vector representation of $SO(2n)$. For this purpose I used the $\gamma_\mu$ basis and tried to find the weights for $A_\mu \gamma^\mu$.

The problem is that certain elements of the cartan algebra just commute with parts of the $A_\mu \gamma^\mu$ sum. For example:

$H_2 (A_1 \gamma^1) = (2 \alpha \gamma_3 \gamma_4) \cdot (A_1 \gamma^1) = (2 \alpha ) \cdot (A_1 \gamma^1) \gamma_3 \gamma_4$

since the commutation of the 2 $\gamma$ gives a factor of $(-1)^2$. Raising and lowering indices is without consequence since the metric for the Clifford algebra is euclidean ( $\delta ^{\mu \nu}$ ).

But this parts should rather go to zero to get a linear action from $H$ on the vector represention.

What is wrong with the approach above? or should it work? Is there a way to justify that the commutation corresponds to a zero element ( or zero weight if it commutes with the wohle $A_\mu \gamma^\mu$)?

• This seems to be a pure math question. – ACuriousMind Feb 5 '16 at 17:16
• The question is related to higher dimensional spacetime lorentz group, where $A_\mu$ vector bosons. – LOQ Feb 5 '16 at 17:27

The problem with this approach is that it mixes up different denotations.

The action of the Cartan subalgebra on the vector representation (or standard representation in math literature) is defined as given above:

$$H_{2}(A_{1}*e_1)$$

by matrix multiplication on a vector. Where $e_1$ is a basis vector of the $R^{2n}$.

However, in the question above we do not act on the vector space $R^{2n}$ but rather on the vector space $R^{n \times n}$. The dimension of the matrix-vector space is reduced by the constraints of the Clifford algebra to have also dimension $2n$.

Nonetheless, we are acting with the Cartan Subalgebra not on the standard vector space but on the vector space of the $n \times n$ matrices. The appropriate action on this vector space is given by the commutator:

$$[H_2, A_1 \gamma^1] = 0$$

Then the weights can be read of in the correct way.