The Category of Representations of a Group
Sorry for missing yesterday. I had this written up but completely forgot to post it while getting prepared for next week’s trip back to a city. Speaking of which, I’ll be heading off for the week, and I’ll just give things here a rest until the beginning of December. Except for the Samples, and maybe an I Made It or so…
Okay, let’s say we have a group 
We start with two representations 
![\mathbb{F}[G]](https://s0.wp.com/latex.php?latex=%5Cmathbb%7BF%7D%5BG%5D&bg=e6e6e6&fg=333333&s=0 "\mathbb{F}[G]")
\right](v\otimes w)=\left[\rho(g)\otimes\sigma(g)\right](v\otimes w)\\=\left[\rho(g)\right](v)\otimes\left[\sigma(g)\right](w)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D%5Cleft%5B%5Cleft%5B%5Crho%5Cotimes%5Csigma%5Cright%5D%28g%29%5Cright%5D%28v%5Cotimes+w%29%3D%5Cleft%5B%5Crho%28g%29%5Cotimes%5Csigma%28g%29%5Cright%5D%28v%5Cotimes+w%29%5C%5C%3D%5Cleft%5B%5Crho%28g%29%5Cright%5D%28v%29%5Cotimes%5Cleft%5B%5Csigma%28g%29%5Cright%5D%28w%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\begin{aligned}\left[\left[\rho\otimes\sigma\right](g)\right](v\otimes w)=\left[\rho(g)\otimes\sigma(g)\right](v\otimes w)\\=\left[\rho(g)\right](v)\otimes\left[\sigma(g)\right](w)\end{aligned}")
That is, we make two copies of the group element 
Symmetry is straightforward. We just use the twist on the underlying vector spaces, and it’s automatically an intertwiner of the actions, so it defines a morphism between the representations.
Duals, though, take a bit of work. Remember that the antipode of 
=\left[\rho(g^{-1})^*\right](\lambda)\\=\lambda\circ\rho(g^{-1})\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D%5Cleft%5B%5Crho%5E%2A%28g%29%5Cright%5D%28%5Clambda%29%3D%5Cleft%5B%5Crho%28g%5E%7B-1%7D%29%5E%2A%5Cright%5D%28%5Clambda%29%5C%5C%3D%5Clambda%5Ccirc%5Crho%28g%5E%7B-1%7D%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\begin{aligned}\left[\rho^*(g)\right](\lambda)=\left[\rho(g^{-1})^*\right](\lambda)\\=\lambda\circ\rho(g^{-1})\end{aligned}")
Composing linear maps from the right reverses the order of multiplication from that in the group, but taking the inverse of
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[…] As with any other group, we have dual representations. That is, we immediately get an action of on . And we’ve seen it already! When we talked about the coevaluation on vector spaces we worked out how a change of basis affects linear functionals. What we found is that if is our action on , then the action on is by the transpose — the dual — of . And this is exactly the dual representation. […]
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