2
votes
Accepted
Representation theoretic condition to show an inner product vanishes
Consider the representation $ (W^* \otimes R \otimes W, \lambda^* \otimes \omega \otimes \lambda) $. The map
$$
W^* \otimes R \otimes W \to \mathbb{C}
$$
given by
$$
w_1 \otimes T \otimes w_2 \mapsto ...
- 7,218
1
vote
Accepted
Homomorphisms $U(1) \to SU(2)$
The matrices in the image commute and thus are simultaneously diagonalizable, so indeed the same $h$ works for all.
- 28.4k
1
vote
Accepted
Show that $\lambda:G \to \mathbb{R}^\ast$ where $R_h^\ast \mu = \lambda (h) \mu$ is smooth
The multiplication map $m\colon G\times G\to G$, $m(g,h) = gh$, is smooth. This means that it induces a smooth bundle map $Dm\colon T(G\times G)\to TG$. In particular, $R_h(g)=m(g,h)$ and so for $X\in\...
- 112k
1
vote
Generators for general / special linear groups
The reference for the 2-element generators mentioned by
brett stevens is
D. E. Taylor:
Pairs of Generators for Matrix Groups. I,
The Cayley Bulletin, 3, 1987, 76-85
https://arxiv.org/abs/2201.09155v1
- 17.6k
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