3.15. Lecture 15

3.15.1. Frobenius Reciprocity

Let $H$ be a subgroup of $G$ and let $(\pi,V)$ be a representation of $G$ . Recall that we have already defined what we mean by the restriction of $(\pi,V)$ to $H$ , which we initially denoted as $(\pi|_{H},V)$ . It is also often denoted as $\operatorname{Res}^{G}_{H}(\pi,G)$ or $(\pi,V)\downarrow_{H}^{G}$ . We will now see how restriction pairs with induction in Frobenius Reciprocity.

Theorem 3.15.1 (Frobenius Reciprocity, 1898).

Let $(\pi,V)$ be a representation of $H\leq G$ . Then for any representation $(\rho,W)$ of $G$ , we have

\operatorname{Hom}_{G}(V_{{\bf r}},W)\cong\operatorname{Hom}_{H}(V,W),

where $(\mathrm{Ind}_{H}^{G}\pi,V_{{\bf r}})$ is the induced representation of $\pi$ , and $(\rho|_{H},W)$ is the restricted representation of $\rho$ .

Proof.

We will create linear maps between these spaces, and show that they are inverses of each other. Firstly, define $\Phi:\operatorname{Hom}_{H}(V,W)\rightarrow\operatorname{Hom}(V_{{\bf r}},W)$ by setting

\Phi(T)\left(\sum_{i}r_{i}{\bf v}_{i}\right):=\sum_{i}\rho(r_{i})T({\bf v}_{i})

for all $T\in\operatorname{Hom}_{H}(V,W)$ and $\sum_{i}r_{i}{\bf v}_{i}\in V_{{\bf r}}$ . On the other hand, WLOG we assume that $e\in r_{1}H$ (hence $r_{1}\in H$ ), and define $\Psi:\operatorname{Hom}_{G}(V_{{\bf r}},W)\rightarrow\operatorname{Hom}(V,W)$ by

\Psi(S){\bf v}:=\rho(r_{1}^{-1})S(r_{1}{\bf v})

for all $S\in\operatorname{Hom}_{G}(V_{{\bf r}},W)$ and ${\bf v}\in V$ .

Since they are defined using evaluation, both $\Phi$ and $\Psi$ are linear, so it remains to verify that:

(i)

$\Phi(T)$ is a $G$ -homomorphism,
(ii)

$\Psi(S)$ is an $H$ -homomorphism,
(iii) & (iv)

they are left/right inverse of each other (iii) .

(i)

For any $g\in G$ , we have

	$\displaystyle\Phi(T)\big{(}\mathrm{Ind}_{H}^{G}\pi(g)r_{i}{\bf v}\big{)}=$	$\displaystyle\Phi(T)\big{(}r_{j(g,i)}\pi(h_{(g,i)}){\bf v}\big{)}=\rho\big{(}r% _{j(g,i)}\big{)}T\big{(}\pi(h_{(g,i)}){\bf v}\big{)}$
		$\displaystyle=\rho\big{(}r_{j(g,i)}\big{)}\rho(h_{(g,i)})T({\bf v})=\rho\big{(% }r_{j(g,i)}h_{(g,i)})T({\bf v})$
		$\displaystyle=\rho(gr_{i})T({\bf v})=\rho(g)\rho(r_{i})T({\bf v})=\rho(g)\Phi(% T)(r_{i}{\bf v}).$

(ii)

Since $r_{1}\in H$ , for any $h\in H$ , we have

	$\displaystyle\Psi(S)\pi(h){\bf v}=$	$\displaystyle\rho(r_{1}^{-1})S(r_{1}\pi(h){\bf v})=\rho(r_{1})^{-1}S\big{(}% \mathrm{Ind}_{H}^{G}\pi(r_{1}hr_{1}^{-1})r_{1}{\bf v}\big{)}$
		$\displaystyle=\rho(r_{1}^{-1})\rho(r_{1}hr_{1}^{-1})S(r_{1}{\bf v})=\rho(h)% \rho(r_{1}^{-1})S(r_{1}{\bf v})=\pi(h)\Psi(S){\bf v}.$

(iii)

For any $r_{i}{\bf v}\in r_{i}V$ ,

$\Phi\big{(}\Psi(S)\big{)}r_{i}{\bf v}=\rho(r_{i})\Psi(S){\bf v}=\rho(r_{i})% \rho(r_{1}^{-1})S(r_{1}{\bf v})=S(\mathrm{Ind}_{H}^{G}\pi(r_{i}r_{1}^{-1})r_{1% }{\bf v})=S(r_{i}{\bf v}),$

hence $\Phi(\Psi(S))=S$ .
(iv)

For any ${\bf v}\in V$ ,

$\Psi(\Phi(T)){\bf v}=\rho(r_{1}^{-1})\Phi(T)(r_{1}{\bf v})=\rho(r_{1}^{-1})(% \rho(r_{1})T({\bf v}))=T{\bf v},$

hence $\Psi(\Phi(T))=T$ .

∎

This has a nice application to characters; we first introduce some convenient notation.

Given a representation $(\pi,V)$ and a class function $f\in CF(G)$ , we write

\mathrm{Ind}_{H}^{G}\chi_{\pi}:=\chi_{\mathrm{Ind}_{H}^{G}\pi},\qquad% \operatorname{Res}_{H}^{G}f:=f|_{H}.

It is also common to see $\chi_{\pi}\uparrow_{H}^{G}$ and $f\downarrow_{H}^{G}$ .

Corollary 3.15.2.

Let $H\leq G$ . For all $f\in CF(G)$ and finite-dimensional representations $(\pi,V)$ of $H$ ,

\langle\mathrm{Ind}_{H}^{G}\chi_{\pi},f\rangle_{G}=\langle\chi_{\pi},% \operatorname{Res}_{H}^{G}f\rangle_{H}.

Remark 3.15.3.

Let $V,W$ be two inner product spaces. Given $A:V\rightarrow W$ , we define $A^{*}:W\rightarrow V$ by the formula

\langle A{\bf v},{\bf w}\rangle_{W}=\langle{\bf v},A^{*}{\bf w}\rangle_{V}% \qquad\forall\;{\bf v}\in V,\;{\bf w}\in W.

The map $A^{*}$ is called the adjoint of $A$ . Because of this, we say that $\mathrm{Ind}_{H}^{G}$ is the adjoint of $\operatorname{Res}_{H}^{G}$ .

Proof.

Since the irreducible characters of $G$ form an orthonormal basis of $CF(G)$ , we may write

f=\sum_{\rho\in\mathrm{Irr}(G)}\langle f,\chi_{\rho}\rangle_{G}\chi_{\rho}.

Then for a representation $\pi$ of $H$ , we have

\displaystyle\langle\mathrm{Ind}_{H}^{G}\chi_{\pi},f\rangle_{G}=\sum_{\rho\in% \mathrm{Irr}(G)}\overline{\langle f,\chi_{\rho}\rangle_{G}}\langle\mathrm{Ind}% _{H}^{G}\chi_{\pi},\chi_{\rho}\rangle_{G}.

By Lemma 2.9.6 and Theorem 3.15.1,

\langle\mathrm{Ind}_{H}^{G}\chi_{\pi},\chi_{\rho}\rangle_{G}=\operatorname{dim% }\big{(}\operatorname{Hom}_{G}(V_{{\bf r}},W_{\rho})\big{)}=\operatorname{dim}% \big{(}\operatorname{Hom}_{H}(V,W_{\rho})\big{)}=\langle\chi_{\pi},% \operatorname{Res}_{H}^{G}\chi_{\rho}\rangle_{H}.

Substituting this into the previous identity gives

\langle\mathrm{Ind}_{H}^{G}\chi_{\pi},f\rangle_{G}=\left\langle\chi_{\pi},\sum% _{\rho\in\mathrm{Irr}(G)}\langle f,\chi_{\rho}\rangle_{G}\operatorname{Res}_{H% }^{G}\chi_{\rho}\right\rangle_{H}=\langle\chi_{\pi},\operatorname{Res}_{H}^{G}% f\rangle_{H}.

∎

Example 3.15.4.

Table 3.2 below shows the irreducible characters $\psi_{i}$ of $S_{4}$ and $\chi_{i}$ of $S_{3}$ .

\begin{array}[]{l|rrrrr}&e&(12)&(12)(34)&(123)&(1234)\\ \hline\cr\psi_{0}&1&1&1&1&1\\ \psi_{1}&1&-1&1&1&-1\\ \psi_{2}&2&0&2&-1&-1\\ \psi_{3}&3&1&-1&0&-1\\ \psi_{4}&3&-1&-1&0&1\\ \hline\cr\hline\cr\chi_{0}&1&1&NA&1&NA\\ \chi_{1}&1&-1&NA&1&NA\\ \chi_{2}&2&0&NA&-1&NA\\ \end{array}

Table 3.2. Characters of

S_{3}

and

S_{4}

We regard $S_{3}$ as the subgroup of $S_{4}$ of elements that fix 4, and use Frobenius reciprocity to compute $\mathrm{Ind}_{S_{3}}^{S_{4}}\chi_{2}$ .

For each $i$ , Frobenius reciprocity implies that

\langle\mathrm{Ind}_{S_{3}}^{S_{4}}\chi_{2},\psi_{i}\rangle_{S_{4}}=\langle% \chi_{2},\psi_{i}|_{S_{3}}\rangle_{S_{3}}.

The right hand side is easily seen to be zero for $i=0,1$ and one for $i=2,3,4$ . We therefore have

\mathrm{Ind}_{S_{3}}^{S_{4}}\chi_{2}=\psi_{2}+\psi_{3}+\psi_{4}.

Corollary 3.15.5.

Let $(\pi,V)$ be a representation of $H\leq G$ . The representation $(\mathrm{Ind}_{H}^{G}\pi,V_{\bf r})$ is independent of the choice of ${\bf r}$ , up to isomorphism.

Proof.

By Corollary 3.15.2 the character of $(\mathrm{Ind}_{H}^{G}\pi,V_{\bf r})$ is entirely determined by the restrictions of irreducible characters of $G$ and the character of $\pi$ . Thus, by Theorem 2.9.7 any choice of ${\bf r}$ gives an isomorphic representation. ∎

3.15.2. Exercises

Problem 82. For each irreducible representation of $S_{4}$ , decompose its induction to $S_{5}$ into irreducibles (where $S_{4}$ is regarded as the subgroup of elements of $S_{5}$ that fix $5\in\{1,\ldots,5\}$ ).

Problem 83. Let $(\pi,V)$ be an irreducible representation of $G$ and $H$ a subgroup of $G$ . Show that $(\pi,V)$ isomorphic to a subrepresentation of a representation induced from an irreducible representation of $H$ .

Problem 84. Let $\chi_{\pi}$ be an irreducible character of $H\leq G$ , and write

\mathrm{Ind}^{G}_{H}\chi_{\pi}=\sum_{\sigma\in\mathrm{Irr}(G)}d_{\sigma}\chi_{% \sigma}.

Show that $\sum_{\sigma}d_{\sigma}^{2}\leq[G:H]$ .

Problem 85. Let $H$ be a subgroup of $G$ . Show

(a)

If $(\pi_{1},V_{1})$ , $(\pi_{2},V_{2})$ are representations of $H$ , then

$\mathrm{Ind}^{G}_{H}\big{(}(\pi_{1},V_{1})\oplus(\pi_{2},V_{2})\big{)}\cong% \mathrm{Ind}^{G}_{H}(\pi_{1},V_{1})\oplus\mathrm{Ind}^{G}_{H}(\pi_{2},V_{2}).$
(b)

If $K\leq H\leq G$ , and $(\rho,U)$ is a representation of $K$ , then

$\mathrm{Ind}_{K}^{G}(\rho,U)\cong\mathrm{Ind}_{H}^{G}\big{(}\mathrm{Ind}_{K}^{% H}(\rho,U)\big{)}.$
(c)

If $(\pi,V)$ is a representation of $G$ , then

$\mathrm{Ind}_{H}^{G}\big{(}\operatorname{Res}_{H}^{G}(\pi,V)\big{)}\cong(\pi,V% )\otimes\mathrm{Ind}^{G}_{H}({\,\mathrm{Id}},{\mathbb{C}}).$