2 votes

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 ...
Ian Gershon Teixeira's user avatar
1 vote

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.
hunter's user avatar
  • 28.4k
1 vote

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\...
Ted Shifrin's user avatar
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
ahulpke's user avatar
  • 17.6k

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