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fishturtle1's latest activity
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fishturtle1
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... and the fact that left multiplication in a group is a bijection ##L_{h^{-1}}\, : \,g \longmapsto h^{-1}g##. Just saying, because...
May 8, 2021
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fishturtle1
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Math Challenge - May 2021
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Proof: We have \begin{align*} \tau(h) \circ \text{Sym}(\varphi) &= \tau(h)\circ \frac{1}{\vert G \vert} \sum_{g \in G} \tau(g) \circ...
May 7, 2021
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fishturtle1
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Yes, but ##\operatorname{Sym}\, : \,\operatorname{Hom}(V,W)\longrightarrow \operatorname{Hom}(V,W)## and we want to show that actually...
May 7, 2021
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fishturtle1
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For ##\varphi \in \text{Hom}_{\mathbb{K}} ((\rho, V), (\tau, W))##, we have ##\tau(h) \text{Sym}(\varphi) = \text{Sym}(\varphi)...
May 7, 2021
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fishturtle1
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Now for the last one: why does ##\operatorname{Sym}(\varphi )## commute with representation matrices?
May 7, 2021
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fishturtle1
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I think the calculation is \begin{align*} \text{Sym}(\varphi) &= \frac{1}{\vert G \vert} \sum_{g \in G} \tau(g) \circ \varphi \circ...
May 7, 2021
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fishturtle1
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(1) ##\operatorname{Im(Sym)} \subseteq \operatorname{Hom}(V,W).## You haven't pointed out that well-definition is one of the 5 claims...
May 7, 2021
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fishturtle1
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Math Challenge - May 2021
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May 7, 2021
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fishturtle1
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Math Challenge - May 2021
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that clears things up, thank you!
May 7, 2021
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fishturtle1
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I don't see why you need the group field, because linearity of homomorphism spaces is all that is used. ##\text{Hom}_{\mathbb{K}}((\rho...
May 7, 2021
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fishturtle1
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Math Challenge - May 2021
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One other question, what is ##\text{Hom}_{\mathbb{K}}((\rho, V), (\tau, W))##? I'm pretty sure ##\text{Hom}_{\mathbb{K}}(V, W)## is the...
May 7, 2021
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fishturtle1
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Yes. As functions, it is the composition, which in coordinates is matrix multiplication.
May 7, 2021
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fishturtle1
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What is the difference?
May 7, 2021
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fishturtle1
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Math Challenge - May 2021
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I think I get it now, I had to look up the definition of GL(V). So, ##\rho(g^{-1})## is an automorphism of ##V## and ##\tau(g)## is an...
May 7, 2021
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fishturtle1
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Sorry for the dumb question but do the ##\circ##'s in 4) mean function composition or matrix multiplication? As I understand it...
May 7, 2021
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