The Unapologetic Mathematician

Mathematics for the interested outsider

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.

November 21, 2008 Posted by | Algebra, Category theory, Group theory, Representation Theory | 1 Comment

   

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