The Unapologetic Mathematician

Left and Right Morphism Spaces

One more explicit parallel between left and right representations: we have morphisms between right $G$-modules just like we had between left $G$-modules. I won’t really go into the details — they’re pretty straightforward — but it’s helpful to see how our notation works.

In the case of left modules, we had a vector space $\hom_G({}_GV,{}_GW)$. Now in the case of right modules we also have a vector space $\hom_G(V_G,W_G)$. We use the same notation $\hom_G$ in both cases, and rely on context to tell us whether we’re talking about right or left module morphisms. In a sense, applying $\hom_G$ “eats up” an action of $G$ on each module, and on the same side.

We can see this even more clearly when we add another action to one of the modules. Let’s say that $V$ carries a left action of $G$ — we write ${}_GV$ — and $W$ carries commuting left actions of both $G$ and another group $H$ — we write ${}_{GH}W$. I say that there is a “residual” left action of $H$ on the space of left $G$-module morphisms. That is, the space $\hom_G({}_GV,{}_{GH}W)$ “eats up” an action of $G$ on each module, and it leaves the left action of $H$ behind.

So, how could $H$ act on the space of morphisms? Well, let $f:V\to W$ be an intertwinor of the $G$ actions, and let any $h\in H$ act on $f$ by sending it to $hf$, defined by $\left[hf\right](v)=hf(v)$. That is, $hf:V\to W$ first sends a vector $v$ to a vector $f(v)$, and then $h$ acts on the left to give a new vector $hf(v)$. We must check that this is an intertwinor, and not just a linear map from $V$ to $W$. For any $g\in G$, we calculate

\displaystyle\begin{aligned}\left[hf\right](gv)&=hf(gv)\\&=hgf(v)\\&=ghf(v)\\&=g\left[hf\right](v)\end{aligned}

using first the fact that $f$ is an intertwinor, and then the fact that the action of $H$ commutes with that of $G$ to pull $g$ all the way out to the left.

Similarly, if $H$ is an extra action on the right of $W$, we have a “residual” right action on the space $\hom_G({}_GV,{}_GW_H)$. And the same goes for right $G$-modules: we have a residual left action of $H$ on $\hom_G(V_G,{}_HW_G)$, and a residual right action on $\hom_G(V_G,W_{GH})$.

It’s a little more complicated when we have extra commuting actions on $V$. The complication is connected to the fact that the hom functor is contravariant in its first argument, which if you don’t know much about category theory you don’t need to care about. The important thing is that if $V$ has an extra left action of $H$, then the space $\hom_G({}_{GH}V,{}_GW)$ will have a residual right action of $H$.

In this case, given a map $f$ intertwining the $G$ actions, we define $fh$ by $\left[fh\right](v)=f(hv)$. We should verify that this is, indeed, a right action:

\displaystyle\begin{aligned}\left[(fh_1)h_2\right](v)&=\left[fh_1\right](h_2v)\\&=f(h_1(h_2v))\\&=f((h_1h_2)v)\\&=\left[f(h_1h_2)\right](v)\end{aligned}

using the fact that $H$ acts on the left on $V$. Again, we must verify that $fh$ is actually another intertwinor:

\displaystyle\begin{aligned}\left[fh\right](gv)&=f(hgv)\\&=f(ghv)\\&=gf(hv)\\&=g\left[fh\right](v)\end{aligned}

using the fact that $f$ is an intertwinor and the actions of $G$ and $H$ on $V$ commute.

Similarly, we find a residual right action on the space $\hom_G({}_HV_G,W_G)$, and residual left actions on the spaces $\hom_G({}_GV_H,{}_GW)$ and $\hom_G(V_{GH},W_G)$.