# Frobenius reciprocity, what values can the restriction of irreducible characters take?

Let $$G$$ be a group and $$H$$ a subgroup. Let $$\psi$$ be a character on $$G$$ and $$\phi$$ be a character on $$H$$. The Frobenius reciprocity tells us that $${\displaystyle \langle \operatorname {Ind} _{H}^{G}\psi ,\phi \rangle _{G}=\langle \psi ,\operatorname {Res} _{H}^{G}\phi \rangle _{H}}$$. Now suppose $$\psi$$ is an irreducible character on $$H$$ and $$\phi$$ is an irreducible character on $$G$$. Does $$\operatorname {Res} _{H}^{G}\phi$$ need to be exactly one of the irreducible representations of $$H$$ or can it be a multiple of it? Or can it even be the sum of different irreducible characters of $$H$$? The latter seem impossible to me but I don't understand why.

• The coefficients can take all values and there can be any number of non-isomorphic constituents. I am only giving this as a comment because I am not providing examples. Commented Jul 29, 2020 at 10:03

Firstly, I think you should have $$\psi$$ is a character of $$H$$ and $$\phi$$ a character of $$G$$ in order for your inductions and restrictions to make sense.
Assume that $$G$$ is finite and $$H$$ is the identity subgroup. Then $$H$$ only has one irreducible representation: the trivial one. Now let $$G$$ be a nonabelian group and take an irreducible representation of dimension greater than $$1$$ (eg the standard one in $$S_3$$). Then the restriction has no choice but to be $$\dim \phi$$ copies of the trivial one when restricted to $$H$$.
If we stick with $$G=S_3$$ and the standard $$2$$-dim representation, then its restriction to any subgroup of order two is the regular representation of $$C_2$$, i.e. the trivial one plus the nontrivial one so different constituents can also occur.
• Thank you. What is the standard 2-dim representation of $S_3$? Commented Jul 29, 2020 at 12:32
• Simply put, its the only 2-dimensional irreducible representation of $S_3$. See groupprops.subwiki.org/wiki/… for the actual representation Commented Jul 29, 2020 at 15:09