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1

The sentence above C.11 explicitly says that they talk about 3-forms under $SO(6)$, i.e. antisymmetric tensors $T_{[abc]}$ where $a,b,c=1,2,3,4,5,6$. Those have $$\frac{6\times 5\times 4}{3\times 2 \times 1} = 20$$ components. By the Dirac matrix calculus, all differential forms may be obtained from the tensor product of two spinors and the Dirac spinor is ...

3

In quantum mechanics, the relevant representations of symmetry groups on the space of states are not our usual linear representation, but projective representations on the Hilbert space. The projective representations of a semi-simple Lie group - such as the rotation group $\mathrm{SO}(n)$ - are in bijection to linear representations of its universal cover. ...

2

The spin group is related to spin-half objects, called spinors. If you rotate a spinor by 360 degrees, you get back the negative of the spinor you started with. Now it would be nice if you could represent the action of this rotation by saying that an element of $SO(n)$ is acting on the spinor. However, this cannot be done because a rotation by 360 degrees is ...

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The issue is that the "spin representation of $SO(3)$" is not a representation of $SO(3)$ at all, but a representation of its double cover $SU(2)$ (see https://en.wikipedia.org/wiki/Spin_group). Since we sometimes write down representations in terms of infinitesimal generators (in other words, as a representation of the Lie algebra of the Lie group in ...

3

In the $SU(4)$ language, the 10-dimensional representation is the symmetric spintensor $T_{(ab)}$ with $4\times 5 / (2\times 1) = 10$ components. In the $SO(6)$ representation, it is the self-dual 3-form with $$\frac 12 \cdot \frac{ 6\times 5 \times 4}{3\times 2 \times 1} = 10$$ components. It's the tensor $T_{[kmn]}$ that also obeys  T_{kmn} = \frac{\...

0

this question is 2 years old, but I thought it's never too late. I'm not sure about the definite answer, but here are my thoughts. Take the SO(6) algebra viewpoint. The $\mathbf{6}$ is the fundamental (vector) representation, and the $\mathbf{4}$ is the spinor representation. So we are looking for symbols $\Sigma_{AB}^I$ that combine two spinors into a ...

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