The Dimension of the Space of Tensors Over the Group Algebra

Now we can return to the space of tensor products over the group algebra and take a more solid pass at calculating its dimension. Key to this approach will be the isomorphism $V\otimes_GW\cong(V\otimes W)^G$ .

First off, we want to calculate the character of $V\otimes W$ . If $V$ — as a left $G$ -module — has character $\chi$ and $W$ has character $\psi$ , then we know that the inner tensor product has character

$\displaystyle\chi\otimes\psi(g)=\chi(g)\psi(g)$

Next, we recall that the submodule of invariants $(V\otimes W)^G$ can be written as

$\displaystyle(V\otimes W)^G\cong V^\mathrm{triv}\otimes\hom_G(V^\mathrm{triv},V\otimes W)$

Now, we know that $\dim(V^\mathrm{triv})=1$ , and thus the dimension of our space of invariants is the dimension of the $\hom$ space. We’ve seen that this is the multiplicity of the trivial representation in $V\otimes W$ , which we’ve also seen is the inner product $\langle\chi^\mathrm{triv},\chi\otimes\psi\rangle$ . We calculate:

$\displaystyle\begin{aligned}\langle\chi^\mathrm{triv},\chi\otimes\psi\rangle&=\frac{1}{\lvert G\rvert}\sum\limits_{g\in G}\overline{\chi^\mathrm{triv}(g)}\chi(g)\psi(g)\\&=\frac{1}{\lvert G\rvert}\sum\limits_{g\in G}\chi(g)\psi(g)\end{aligned}$

This may not be as straghtforward and generic a result as the last one, but it’s at least easily calculated for any given pair of modules $V$ and $W$ .

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This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.

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The Unapologetic Mathematician

Mathematics for the interested outsider