Induced Representations, Frobenius Reciprocity and Applications — Examples

Since $[S_3:A_3]=2$, Frobenius' formula [F1] at the identity gives ${\mathrm{Ind}}_{A_3}^{S_3}\theta (e)=2$. If $g$ is a transposition, no conjugate of $g$ lies in $A_3$, so Frobenius' formula gives ${\mathrm{Ind}}_{A_3}^{S_3}\theta (g)=0$.

If $g$ is a $3$-cycle, then $A_3$ is normal in $S_3$, so every $x\in S_3$ satisfies $x^{-1}gx\in A_3$. Exactly three of those conjugates equal $(123)$ and three equal $(132)$, so [F1] gives ${\mathrm{Ind}}_{A_3}^{S_3}\theta (g)=(3\omega +3{\omega }^2)\mathrm{/}3=\omega +{\omega }^2=-1$.

Therefore the induced character has values $(2,0,-1)$ on the three class types of $S_3$. Its self-inner-product is $(1\mathrm{/}6)(2^2+3\cdot 0^2+2\cdot (-1{)}^2)=1$, so [F2] makes it irreducible; the value at $e$ shows that its degree is $2$.

Restricting the values $(2,0,-1)$ from [F1] to $A_3=\{e,(123),(132)\}$ gives ${\mathrm{Res}}_{A_3}^{S_3}{\chi }_2=(2,-1,-1)$.

Frobenius reciprocity [F2] gives $⟨\theta ,{\mathrm{Res}}_{A_3}^{S_3}{\chi }_2{⟩}_{A_3}=⟨{\mathrm{Ind}}_{A_3}^{S_3}\theta ,{\chi }_2{⟩}_{S_3}=⟨{\chi }_2,{\chi }_2{⟩}_{S_3}=1$, and the same computation with $\bar{\theta }$ gives multiplicity $1$ for $\bar{\theta }$.

The restriction has degree $2$ at the identity, while the two nontrivial linear characters already account for degree $1+1=2$. Hence no trivial summand occurs, and ${\mathrm{Res}}_{A_3}^{S_3}{\chi }_2=\theta +\bar{\theta }$.

By [F1] and [F2], the induced character counts fixed cosets of the left action on the three cosets of $H$. The identity fixes all three cosets, a transposition fixes exactly one coset, and a $3$-cycle fixes none, so ${\mathrm{Ind}}_H^{S_3}{\mathbf{1}}_H=(3,1,0)$.

Subtracting the trivial character $(1,1,1)$ gives the class function $(2,0,-1)$. Its self-inner-product is $(1\mathrm{/}6)(2^2+3\cdot 0^2+2\cdot (-1{)}^2)=1$, so [F3] makes it irreducible; its value at the identity is $2$, so it has degree $2$.

Therefore ${\mathrm{Ind}}_H^{S_3}{\mathbf{1}}_H=1+{\chi }_2$ with ${\chi }_2$ irreducible of degree $2$.

By [F1], $⟨{\mathrm{Ind}}_{A_3}^{S_3}\theta ,{\chi }_2{⟩}_{S_3}=⟨{\chi }_2,{\chi }_2{⟩}_{S_3}=1$. By [F3], the matching inner product on $A_3$ is therefore also $1$.

By [F2], $⟨{\mathrm{Ind}}_{⟨(12)⟩}^{S_3}\mathbf{1},{\chi }_2{⟩}_{S_3}=⟨1+{\chi }_2,{\chi }_2{⟩}_{S_3}=1$. Again [F3] makes the inner product after restriction equal to the same value.

So both preceding inductions realize Frobenius reciprocity numerically with multiplicity $1$ on each side.

By [F1], $C_4$ is a finite cyclic group of order $4$, hence finite abelian.

Over $\mathbb{C}$, every irreducible representation of the finite abelian group $C_4$ has degree $1$ by [F2]. Therefore no irreducible complex character of $C_4$ has degree $2$, even though $2\mid 4$.

This shows that [F3] is not an equivalence: a divisor of $\mathrm{\mid }G\mathrm{\mid }$ need not occur as an irreducible degree.

The trivial character ${\mathbf{1}}_H$ of the order-two subgroup $H$ is irreducible because every one-dimensional character is irreducible.

But [F1] shows that its induction to $S_3$ is $1+{\chi }_2$, a nontrivial sum of two characters. So the induced character is reducible.

This single witness refutes the claim that induced irreducible characters are always irreducible.

By [F1], ${\mathrm{Ind}}_{A_3}^{S_3}\theta ={\chi }_2$.

Applying restriction and then [F2] gives ${\mathrm{Res}}_{A_3}^{S_3}{\mathrm{Ind}}_{A_3}^{S_3}\theta ={\mathrm{Res}}_{A_3}^{S_3}{\chi }_2=\theta +\bar{\theta }\mathrm{\ne }\theta$.

So induction followed by restriction is not the identity in general.

The character ${\chi }_2$ is irreducible on $S_3$ by the example that constructs it.

But [F1] writes its restriction to $A_3$ as the sum of two distinct nontrivial characters, so the restricted representation is reducible.

Therefore restriction does not preserve irreducibility in general.

The integer $2$ divides $\mathrm{\mid }{C}_4\mathrm{\mid }=4$.

But [F1] shows that no irreducible complex character of $C_4$ has degree $2$.

Hence not every divisor of the group order occurs as an irreducible character degree.