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One reason there are more possible eigenvalues of the Casimir operator of the rotations than appear in the spherical harmonics is that the spherical harmonics are proper representations of $\mathrm{SO}(n)$ while the possible values for the Casimir operator classify the possible irreducible representations of $\mathfrak{so}(n)$. By general Lie theoretic ...

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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 ...

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A 3D cube with pacman topology is translationally invariant and not rotationally invariant. A space like this is a possible (but unlikely) flat space part of a cosmological spacetime

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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. ...

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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 ...

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The quotient $G/H$ is a system of cosets so that for $h~\in~H$ and $g$ a set of group elements in the $gh~\in~H$ for left cosets and $hg~\in~H$. We then have $gH~=~Hg$ or that the subgroup $H$ is a normal subgroup. So for $SU(n)/U(1)$ we may look at $SU(2)$ and $SU(3)$. The Pauli matrices $\sigma_i,~i~=~1,2,3$ generate transformations of the form $$g_i~=... 5 The group U(N)\times U(N) acts on your original space, but some group elements act in the same way: the action is not faithful, in other words, this groups maps onto the symmetry group (this is implicitly assumed by what you wrote), but not injectively. Specifically, the diagonal subgroup of scalar matrices (i.e. matrices that are scalar multiples of the ... 2 It seems OP's main question is how to understand the representation of the matter fields of YM theory. The matter fields can in principle transform in any representation \rho:G\to {\rm End}(V) of the local gauge group G=SU(N), e.g. the fundamental, or adjoint representation. Here {\rm End}(V) denotes the algebra of endomorphisms on the vector space ... 0 Let S be the stabilizer of some n-qubit code, with r generators (what you call the size of S). The normalizer N(S) is a group of logical operations on the code space. Given some input code state |\psi\rangle, the state N|\psi\rangle, N\in N(S) is also in the code space. Consider that all the elements of the coset nS, n\in N(S) act on the ... 3 The Poincaré group is the semi-direct product of the six-dimensional Lorentz group and the four-dimensional translations and hence ten-dimensional (or "has ten parameters" is less precise diction). Since in a global inertial coordinate system you have to have the Minkowski metric by definition, only those transformations (diffeomorphisms) which preserve the ... 3 Representation of SU(2) is pseudo-real. Which means, if \mathbf{[2]} and \mathbf{[\bar{2}]} are the fundamental and anti-fundamental representation of SU(2), then there exists an anti-symmetric matrix \cal{C}, which connect both of them, as \cal{C}\mathbf{[2]}\cal{C}^{-1}=\mathbf{[\bar{2}]}. Another way of saying this, both \mathbf{[2]} and \... 3 In the SU(4) language, the 10-dimensional representation is the symmetric spintensor T_{(ab)} with 4\times 5 / (2\times 1) = 10 components. In the SO(6) representation, it is the self-dual 3-form with$$ \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{\...

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This question appears to cover a fair amount of territory. I have asked a number of these types of questions. This seems to be asking what is the distinction between the say the s quark and the c quark in their doublet. The question of “why $SU(3)$” is another question, which in some ways includes the question of why there are 3 families of quarks. The ...

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