# Questions tagged [representation-theory]

The systematic study of group representations, which describe abstract groups in terms of linear transformations of vector spaces, such that group elements or their generators are represented as matrices, reducing group-theoretic problems to linear-algebraic ones.

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### Relation of Wigner $d$-matrix $d^l_{m',m} = d^l_{-m,-m'}$

How do you derive the symmetry relation of the Wigner $d$-matrix, i.e., $$d^l_{m',m} = d^l_{-m,-m'}$$ I know how Wikipedia proves this using the fact that $(Y_l^m)^* = (-1)^m Y_l^{-m}$ (basically ...
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### Angular momentum and rotation group representations

In Sakurai's book it's written that the operator $D_{m',m}^{(j)}=\left\langle{j,m'}\Big|\exp{\frac{-i \mathbf{ J\cdot \hat{n} } \phi}{\hbar}}\Big|{j,m}\right\rangle$ is the "$2j+1$-dimensional ...
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### Spin operators in QM

In a text (Introduction to Quantum Mechanics by Griffiths) I am using it states without motivation that spin angular momentum has the same commutations relations as orbital angular momentum (these ...
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### Arriving at the $\big(\pi_\ell,P_\ell(\mathbb{C}^2)\big)$ representation of $\mathfrak{su}(2)$

I think I'm really close, but confused on applying the multivariable chain rule and untangling the result. The $(\Pi_\ell,P_\ell(\mathbb{C}^2))$ representation of $SU(2)$ induced from the fundamental ...
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### Symmetry of Clebsch-Gordan coefficients

The symmetry of clebsch-gordan coefficients $\left< j_1j_2;m_1m_2 \middle| j_1j_2;JM \right>$ under exchange of $j_1,m_1$ and $j_2,m_2$ is \begin{equation} \left< j_1j_2;m_1m_2 \middle| ...
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### What does $\Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu$ mean?

\begin{equation} \Lambda^{-1}_{\frac{1}{2}}\gamma^\mu\Lambda_{\frac{1}{2}}=\Lambda^\mu_{\phantom{\mu}\nu}\gamma^\nu \end{equation} In P&S, p. 42: Equation (3.29) says that the $\gamma$ ...
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### Operator-valued vectors and representation theory

Let $G$ be a Lie group and $\pi : G\to GL(V)$ a finite-dimensional representation of $G$ in the vector space $V$. For every $g\in G$ we have a linear transformation $\pi(g) : V\to V$. Being linear, if ...
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### Representing $su(2)$ Lie algebra on a torus

I've recently taken up the study of QFT (as a post retirement hobby), based on texts by David Tong and Anthony Zee. My question is based on the Lie Algebra of the $SU(2)$ group, and how this may ...
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### Behavior of derivative on Lorentz transformations of spinors

I'm currently working through Supersymmetry Demystified by Patrick Labelle and one passage in particular confuses me. Specifically, if $\eta$ and $\chi$ are right and left Weyl spinors respectively, ...
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### Are all representations of a finite group unitary?

I am reading Zee's Group theory in a nutshell for physicists and came across the following theorem (Page 96): Unitary representations The all-important unitarity theorem states that finite ...
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### How do I show that the tensor product of $\mathbf{3} \otimes \mathbf{\bar{3}}=\mathbf{1} \oplus \mathbf{8}$? [duplicate]

It's often stated that the tensor product of the representations of $SU(3)$ satisfies $\mathbf{3} \otimes \mathbf{\bar{3}}=\mathbf{1} \oplus \mathbf{8}$, and that this implies that if flavour $SU(3)$ ...
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### Representation under which Pauli matrices transform

In Peskin and Schroeder's Quantum Field Theory, there is an identity of Pauli matrices which is connected to the Fierz identity, (equation 3.77) (\sigma^{\mu})_{\alpha\beta}(\sigma_\mu)_{\gamma\...