# The Unapologetic Mathematician

## (Fake) Frobenius Reciprocity

Today, we can prove the Frobenius’ reciprocity formula, which relates induced characters to restricted ones.

Now, naïvely we might hope that induction and restriction would be inverse processes. But this is clearly impossible, since if we start with a $G$-module $V$ with dimension $d$, it restricts to an $H$-module $V\!\!\downarrow^G_H$ which also has dimension $d$. Then we can induce it to a $G$-module $V\!\!\downarrow^G_H\uparrow_H^G$ with dimension $d\tfrac{\lvert G\rvert}{\lvert H\rvert}$. This can’t be the original representation unless $H=G$, which is a pretty trivial example indeed.

So, instead we have the following “reciprocity” relation. If $\chi$ is a character of the group $G$ and $\psi$ is a character of the subgroup $H$, we find that

$\displaystyle\langle\chi\!\!\downarrow^G_H,\psi\rangle_H=\langle\chi,\psi\!\!\uparrow_H^G\rangle_G$

Where the left inner product is that of class functions on $H$, while the right is that of class functions on $G$. We calculate the inner products using our formula

\displaystyle\begin{aligned}\langle\chi,\psi\!\!\uparrow_H^G\rangle_G&=\frac{1}{\lvert G\rvert}\sum\limits_{g\in G}\chi(g^{-1})\psi\!\!\uparrow_H^G(g)\\&=\frac{1}{\lvert G\rvert}\frac{1}{\lvert H\rvert}\sum\limits_{g\in G}\sum\limits_{x\in G}\chi(g^{-1})\psi(x^{-1}gx)\\&=\frac{1}{\lvert G\rvert}\frac{1}{\lvert H\rvert}\sum\limits_{x\in G}\sum\limits_{y\in G}\chi(xy^{-1}x^{-1})\psi(y)\\&=\frac{1}{\lvert G\rvert}\frac{1}{\lvert H\rvert}\sum\limits_{x\in G}\sum\limits_{y\in G}\chi(y^{-1})\psi(y)\\&=\frac{1}{\lvert H\rvert}\sum\limits_{y\in G}\chi(y^{-1})\psi(y)\\&=\frac{1}{\lvert H\rvert}\sum\limits_{y\in H}\chi(y^{-1})\psi(y)\\&=\frac{1}{\lvert H\rvert}\sum\limits_{y\in H}\chi\!\!\downarrow^G_H(y^{-1})\psi(y)\\&=\langle\chi\!\!\downarrow^G_H,\psi\rangle_H\end{aligned}

where we have also used the fact that $\chi$ is a class function on $G$, and that $\psi$ is defined to be zero away from $H$.

As a special case, let $\chi^{(i)}$ and $\chi^{(j)}$ be irreducible characters of $G$ and $H$ respectively, so the inner products are multiplicities. For example,

$\displaystyle\langle\chi^{(i)},\chi^{(j)}\!\!\uparrow_H^G\rangle_G=m_1$

is the multiplicity of $\chi^{(i)}$ in the representation obtained by inducing $\chi^{(j)}$ to a representation of $G$. On the other hand,

$\displaystyle\langle\chi^{(i)}\!\!\downarrow^G_H,\chi^{(j)}\rangle_H=\overline{\langle\chi^{(j)},\chi^{(i)}\!\!\downarrow^G_H\rangle_H}=\overline{m_2}=m_2$

is the multiplicity of $\chi^{(j)}$ in the representation obtained by restricting $\chi^{(i)}$ down to $H$. The Frobenius reciprocity theorem asserts that these multiplicities are identical.

Now, why did I call this post “fake” Frobenius reciprocity? Well, this formula gets a lot of press. But really it’s a pale shadow of the real Frobenius reciprocity theorem. This one is a simple equation that holds at the level of characters, while the real one is a natural isomorphism that holds at the level of representations themselves.

November 30, 2010 -