# The Unapologetic Mathematician

## 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 $G$. This gives us a cocommutative Hopf algebra. Thus the category of representations of $G$ is monoidal — symmetric, even — and has duals. Let’s consider these structures a bit more closely.

We start with two representations $\rho:G\rightarrow\mathrm{GL}(V)$ and $\sigma:G\rightarrow\mathrm{GL}(W)$. We use the comultiplication on $\mathbb{F}[G]$ to give us an action on the tensor product $V\otimes W$. Specifically, we find

\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 $g$, use $\rho$ to act on the first tensorand, and use $\sigma$ to act on the second tensorand. If $\rho$ and $\sigma$ came from actions of $G$ on sets, then this is just what you’d expect from linearizing the product of the $G$-actions.

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 $\mathbb{F}[G]$ sends group elements to their inverses. So if we start with a representation $\rho:G\rightarrow\mathrm{GL}(V)$ we calculate its dual representation on $V^*$:

\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 $g$ reverses it again, and so we have a proper action again.