Frobenius reciprocity

Frobenius reciprocity is a term from the mathematical field of representation theory. He relates induced representations and the restriction of representations to one another.

The Frobenius reciprocity tells us on the one hand that the images and are adjoint to one another. On the other hand, if we consider with an irreducible representation of and be an irreducible representation of then we also get with Frobenius reciprocity that as often is contained in as in ${\displaystyle {\text{Res}}}$ ${\displaystyle {\text{Ind}}}$ ${\displaystyle W}$ ${\displaystyle H}$ ${\displaystyle V}$ ${\displaystyle G,}$ ${\displaystyle W}$ ${\displaystyle {\text{Res}}(V)}$ ${\displaystyle {\text{Ind}}(W)}$ ${\displaystyle V.}$

It is named after Ferdinand Georg Frobenius .

notation

By means of the restriction (engl .: restriction ) can be obtained from a representation of a group of a representation of a subset obtained. Conversely, from a given representation of a subgroup, the so-called induced representation of the whole group can be obtained. ${\displaystyle \phi }$ ${\displaystyle G}$ ${\displaystyle {\text{Res}}(\phi )}$ ${\displaystyle H}$ ${\displaystyle \psi }$ ${\displaystyle H}$ ${\displaystyle {\text{Ind}}(\psi )}$ ${\displaystyle G}$

For representations and their characters but also more generally for class features one is scalar defined. The general form of Frobenius reciprocity uses the scalar product of class functions.

Frobenius reciprocity

Let be a finite group and a subgroup . Let be class functions, then ${\displaystyle G}$ ${\displaystyle H\subset G}$ ${\displaystyle \psi \in \mathbb {C} _{\text{class}}(H)}$ ${\displaystyle \varphi \in \mathbb {C} _{\text{class}}(G)}$

${\displaystyle \langle \psi ,{\text{Res}}\varphi \rangle _{H}=\langle {\text{Ind}}\psi ,\varphi \rangle _{G}}$

The statement applies in particular to the scalar product of characters in representations.

proof

Since every class function can be written as a linear combination of irreducible characters , and is a bilinear form , we can assume an irreducible representation of in or of in without restriction or as a character . We set for Then: ${\displaystyle \langle \cdot ,\cdot \rangle }$ ${\displaystyle \psi }$ ${\displaystyle \varphi }$ ${\displaystyle H}$ ${\displaystyle W}$ ${\displaystyle G}$ ${\displaystyle V}$ ${\displaystyle \psi (s)=0}$ ${\displaystyle s\in G\setminus H.}$

${\displaystyle {\begin{aligned}\langle {\text{Ind}}(\psi ),\varphi \rangle _{G}&={\frac {1}{|G|}}\sum _{t\in G}{\text{Ind}}(\psi )(t)\varphi (t^{-1})\\&={\frac {1}{|G|}}\sum _{t\in G}{\frac {1}{|H|}}\sum _{s\in G \atop s^{-1}ts\in H}\psi (s^{-1}ts)\varphi (t^{-1})\\&={\frac {1}{|G|}}{\frac {1}{|H|}}\sum _{t\in G}\sum _{s\in G}\psi (s^{-1}ts)\varphi ((s^{-1}ts)^{-1})\\&={\frac {1}{|G|}}{\frac {1}{|H|}}\sum _{t\in G}\sum _{s\in G}\psi (t)\varphi (t^{-1})\\&={\frac {1}{|H|}}\sum _{t\in G}\psi (t)\varphi (t^{-1})\\&={\frac {1}{|H|}}\sum _{t\in H}\psi (t)\varphi (t^{-1})\\&={\frac {1}{|H|}}\sum _{t\in H}\psi (t){\text{Res}}(\varphi )(t^{-1})\\&=\langle \psi ,{\text{Res}}(\varphi )\rangle _{H}\end{aligned}}}$

We only used the definition of induction on class functions and exploited the properties of the characters . ${\displaystyle \Box }$

Alternative proof

In the alternative description of the induced representation via group algebra, Frobenius reciprocity is a special case of the equation for the change between rings:

${\displaystyle {\text{Hom}}_{\mathbb {C} [H]}(W,U)={\text{Hom}}_{\mathbb {C} [G]}(\mathbb {C} [G]\otimes _{\mathbb {C} [H]}W,U).}$

This equation is by definition equivalent to

${\displaystyle \langle W,{\text{Res}}(U)\rangle _{H}=\langle W,U\rangle _{H}=\langle {\text{Ind}}(W),U\rangle _{G}.}$

And since this bilinear form agrees with the bilinear form on the corresponding characters, the sentence follows without any recalculation. ${\displaystyle \Box }$

Frobenius reciprocity for compact groups

The Frobenius reciprocity is carried over to compact groups with the modified definition of the scalar product and the bilinear form , whereby the theorem applies here to quadratically integrable functions instead of to class functions and the subgroup must be closed. ${\displaystyle G}$ ${\displaystyle H}$

Web links

• Frobenius reciprocity (nLab)