Decomposing products of spinor representations into anti-symmetric tensors

There is always a natural $2^{[\frac{d}{2}]}$ dimensional spinorial representation of $SO(d-1,1)$ (..induced from a representation of the related Clifford algebra..) and if $[m]$ denote the space of rank-$m$ anti-symmetric tensors in $d$ dimensions.

Then for $d$ even it apparently holds that,

$$2^{\frac{d}{2}} \otimes 2^{\frac{d}{2}} = \bigoplus _ {m=1} ^ {{[\frac{d}{2}]}} [m]$$

• Why is the above true?

..I would think that the dimension of $[m]$ is $^dC_m$ and then even the dimension of the RHS doesn't seem to be $2^{d}$ as it is for the LHS..

• Is the RHS of the above somehow again irreducible representations of $SO(d-1,1)$ or $SO(d)$?

• Because of a duality relation between tensors of rank $m$ and $d-m$ for odd d apparently the above is modified to,

$$2^{[\frac{d}{2}]} \otimes 2^{[\frac{d}{2}]} = \bigoplus_{m=1}^{d} [m] = \bigoplus_{m=1}^{[\frac{d}{2}]-1} [m]^2 + \left[\left[\frac{d}{2}\right]\right].$$

It would be great if someone can explain/help prove/reference the above.

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