14 votes

Why does group representation theory look linear?

A group in itself is not necessarily related to linear spaces. Representation theory (which is a subfield of group theory) is the study of the way in which groups act linearly on vector spaces : a ...
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11 votes

Why does group representation theory look linear?

A group $G$ is an algebraic structure which encodes one pattern of composition of objects. This makes abstract the notion of transformations which may be composed to give new ones. The whole point, ...
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8 votes
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Basic Facts about Lie Algebras

Yes, OP is right: P&S do not distinguish between the abstract Lie group and its defining representation, which is usually the fundamental representation.
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5 votes

Why does group representation theory look linear?

Concerning OP's title question Why does group representation theory look linear? it should be mentioned that there exist in physics (e.g. in the areas of spontaneous symmetry breaking and ...
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5 votes

Help needed in understanding $2\otimes 2=1\oplus 1\oplus 2$ of $SO(2)$

You can parameterize a general $2\times 2$ matrix of real numbers in terms of 4 real numbers $t$, $a$, $s_1$, and $s_2$ as \begin{equation} A = \left( \begin{array} _ \frac{t}{2} + s_1 & s_2 + a \\...
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4 votes

Basic Facts about Lie Algebras

You learnt that in your group theory course, that sounds right. But you must consider the topology. Lie groups are manifolds, so they have tangent spaces (this is an additional structure to being a ...
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4 votes
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Help needed in understanding $2\otimes 2=1\oplus 1\oplus 2$ of $SO(2)$

The "2" is the space of all trace-free symmetric rank-2 tensors. So an arbitrary tensor product of two vectors can be decomposed as $$ A_i B_j = \frac{1}{2} \delta_{ij} A_k B_k + \frac{1}{2}...
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4 votes
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Invariants of inner product in pseudoreal representation of $SU(2)$

P&S are talking about the spinor/defining/fundamental representation $$\eta,\xi~\in~ V~\cong~ \mathbb{C}^2$$ of $SU(2)$. The expression $\epsilon^{\alpha\beta}\eta_{\alpha}\xi_{\beta}$ is $SU(2)$-...
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3 votes
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Find commutator $[P_\mu,K_\nu]$ in conformal group

It is best to work with generators rather than group elements. The generators have a representation in real space, that is $P_\mu = -i\partial_\mu, D = -i x^{\mu} \partial_\mu, M_{\mu\nu} = i(x^{\mu}\...
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3 votes

Help needed in understanding $2\otimes 2=1\oplus 1\oplus 2$ of $SO(2)$

Any $2\times2$ matrix (4 dof) can be broken up into its trace part (1 dof), an antisymmetric part (1 dof) and a symmetric traceless part (2 dof). Hence, $2\times 2=4=1+1+2$.
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2 votes
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What were the $r$ and $n$ of $\theta$s (Polchinski String theory section 8.6 page 265)?

The numbers $r_i\in\mathbb{N}$ are the degeneracies/multiplicies of the eigenvalues $\theta_i$ of the diagonal background gauge field $A_{25}$ in the compactified 25th direction $$ A_{25}~=~-\frac{1}{...
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2 votes

Invariants of inner product in pseudoreal representation of $SU(2)$

I would like add to Qmechanic's excellent answer a bit of context from the practical point of view. Apparently the question arises in classical Quantum mechanics not accounting for relativity. So then ...
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2 votes

Proving rotation about non-intersecting axes leads to translation

The issue is that the composition of two rotations about non-intersecting axes from a translation iff the axes are parallel and the rotations' angles are opposite. In general, their composition will ...
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  • 1,647
2 votes

Can the $SU(3)$ gauge field be put in geometric algebra terms?

The 6D geometric algebra bivectors span $$spin(6)$$ that is isomorphic to Pati-Salam's $$SU(4)$$ which in turn contains $$SU(3)_{color} * U(1)_{B-L}$$
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2 votes
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Questions regarding the 'adjoint of $SO(N)$' section in Zee's group theory

I don't have that text, but your question is standard stuff. I suspect what you are missing is the fast interplay of D-dimensional vectors and the D×D matrices acting on them, sometimes regarded as ...
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1 vote

Clebsch-Gordan Series and Rotation Matrices

The rotation operator \begin{align} D(R)=e^{i\vec \omega\cdot \vec L}& =e^{i\vec \omega \cdot(\vec L^{(1)}+\vec L^{(2)})} = e^{i\vec \omega \cdot\vec L^{(1)}}e^{i\vec \omega \cdot\vec L^{(2)}}\, ,\...
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1 vote

A relation for adjoint representation of $U(N)$ acting on product of matrices and $SU(2)$ generators

Here is a partial remark on a special case that might help you get started on your problem, where everything is completely calculable directly. Take the case of N=2, where you might as well normalize ...
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1 vote
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Combination of 3 spin $\frac{1}{2}$ particles to yield a state of net spin $\frac{1}{2}$

I guess the question I ask can be explained in a simpler manner eh? The narrow question can be answered by elementary linear algebra, of course, but the point of such questions is that you understand ...
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1 vote
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Do $Y$ bosons gain a little bit of mass from Higgs of $\bf 5$ rep in $SU(5)$ GUT theory?

I'll be schematic and cavalier with normalizations and factors... You can do this right in your peculiar conventions. I'll call the v.e.v., of the 24 Higgs v, following the obligatory sourcebook of ...
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1 vote
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Constructing gauge invariants

I am using this reference, Spin Multiplicities, T Curtright, T van Kortryk, and C Zachos, Phys Lett A381 (2017) 422-427. The character of a spin j, dimension 2j+1 irrep of SU(2) is $$ \chi_j (\theta)...
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1 vote

Why does group representation theory look linear?

Linearity is no restriction in the following sense: Every compact Lie group $G$ (in particular every finite group) has a faithful finite-dimensional linear representation, see the Peter-Weyl-theorem. ...
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1 vote

Mathematical characterisation of diffeomorphisms in General Relativity

GR is diffeomorphism invariant. If the spacetime manifold $M$ has a metric $g$, then it is possible that its diffeomorphism group $\mathrm{Diff}(M)$ has a subgroup which preserves its metric. This is ...
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