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Results tagged with lie-groups
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user 104213
Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
6
votes
1
answer
329
views
Single sum of squares of Clebsch–Gordan coefficients
Let $C^{j_3 m_3}_{j_1 m_1 j_2 m_2}$ be the standard Clebsch–Gordan coefficients of $\operatorname{SU}(2)$. They obey the orthogonality relation
$$ \sum_{j_3} \sum_{m_3} \left(C^{j_3 m_3}_{j_1 m_1 j_2 …
- 914
8
votes
0
answers
661
views
Convolution theorem on a non-abelian Lie group
Let $\mathrm{G}$ be a compact (simple, if it helps) non-abelian Lie group and let $\hat{\mathrm{G}}$ be its unitary dual of (equivalence classes) of irreducible unitary representations. Defining the F …
- 914
5
votes
1
answer
2k
views
Clebsch–Gordan decomposition for $\mathrm{SU}(2)$, in indices
Let $\pi_m$, $m \geq 0$, be the unitary irreps of $\mathrm{SU}(2)$. The Clebsch–Gordan decomposition then gives that
$$ \pi_m \otimes \pi_n = \bigoplus_{k=0}^{\min(m,n)}\pi_{m+n-2k}.$$
But suppose I w …
- 914