The Unapologetic Mathematician

Mathematics for the interested outsider

(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 Posted by | Algebra, Group theory, Representation Theory | 3 Comments

   

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