3.13. Lecture 13

3.13.1. Inflation

We now turn to group-theoretic methods to create new representations. Recall that if $N<G$ is a normal subgroup, the quotient $\widetilde{G}=G/N$ is a group with multiplication

g_{1}N\cdot g_{2}N=g_{1}g_{2}N\qquad\forall\;g_{1}N,\;g_{2}N\in\widetilde{G}.

We will show how one can create representations of $G$ out of representations of the smaller group $\widetilde{G}$ .

Definition 3.13.1.

Let $(\pi,V)$ be a representation of $\widetilde{G}=G/N$ . The inflation of $\pi$ to $G$ is denoted $(\widetilde{\pi},V)$ , and defined by

\widetilde{\pi}(g):=\pi(gN)

for all $g\in G$ .

This is also called the lift of the representation from $\widetilde{G}$ to $G$ .

Proposition 3.13.2.

A representation $(\rho,V)$ of $G$ may be expressed as the inflation of a representation of $\widetilde{G}=G/N$ if and only if $N\subseteq\ker\rho$ .

Proof.

Let $(\widetilde{\pi},V)$ be a representation of $G$ such that it is the inflation of a representation $(\pi,V)$ of $\widetilde{G}$ . Then for any $n\in N$ , we have

\widetilde{\pi}(n)=\pi(nN)=\pi(eN)={\,\mathrm{Id}},

so $N\subseteq\ker\widetilde{\pi}$ .

On the other hand, if $N\subseteq\ker\rho$ , then we define a representation $(\pi,V)$ of $\widetilde{G}$ by

\pi(gN)=\rho(g),

for any $gN\in\widetilde{G}$ . This is well-defined, since $\rho(gn)=\rho(g)\rho(n)=\rho(g)$ for all $g\in G$ , $n\in N$ . It is a homomorphism since

\pi(g_{1}N\cdot g_{2}N)=\pi(g_{1}g_{2}N)=\rho(g_{1}g_{2})=\rho(g_{1})\pi(g_{2}% )=\pi(g_{1}N)\pi(g_{2}N),

for all $g_{1}N$ , $g_{2}N$ in $\widetilde{G}$ . Finally, letting $\widetilde{\pi}$ be the lift of $\pi$ to $G$ , we have

\widetilde{\pi}(g)=\pi(gN)=\rho(g),

showing that $\rho$ is indeed the lift of a representation. ∎

Proposition 3.13.3.

Let $(\widetilde{\pi},V)$ be a representation of $G$ that is the inflation of a representation $(\pi,V)$ of $\widetilde{G}$ . Then $(\widetilde{\pi},V)$ is irreducible if and only if $(\pi,V)$ is.

Proof.

If $W$ is an invariant subspace of $V$ for $\pi$ , then for any $g\in G$ ,

\widetilde{\pi}(g)W=\pi(gN)W=W,

and conversely, if $W$ is an invariant subspace for for $\widetilde{\pi}$ , then

\pi(gN)W=\widetilde{\pi}(g)W=W.

The representations thus have subrepresentations in common, and thus are both irreducible or decomposable together. ∎

Example 3.13.4.

$D_{n}/\langle r\rangle=e\langle r\rangle\sqcup s\langle r\rangle\cong C_{2}={% \mathbb{Z}}/2{\mathbb{Z}}$ . There are two irreducible representations of $C_{2}$ , given by the choices $\pi(s\langle r\rangle)=\pm 1$ . Inflating these to $D_{n}$ gives the two 1-dimensional representations of $D_{n}$ with $\widetilde{\pi}(r)=1$ (i.e. the trivial representation and the representation $\epsilon$ : $\epsilon(r)=1$ , $\epsilon(s)=-1$ ).

Recall that the kernel of any group homomorphism is a normal subgroup. By the previous proposition, we may thus realise any representation $\pi$ as the inflation of a representation of $G/\ker\pi$ . The representations that are not inflations are said to be faithful:

Definition 3.13.5.

Let $(\pi,V)$ be a representation of a group $G$ . The representation $\pi$ is said to be faithful if the kernel is trivial, i.e.

\ker\pi=\{g\in G\,|\,\pi(g)={\,\mathrm{Id}}\}=\{e\}.

Remark 3.13.6.

Recall that by Problem 2.7.2(ii), $\ker\pi=\{g\in G\,|\,\chi_{\pi}(g)=\operatorname{dim}(\pi)\}$ .

Thus, every non-trivial irreducible representation of a simple group is faithful. The follow proposition provides a converse to this fact:

Proposition 3.13.7.

Every normal subgroup $N$ of a finite group $G$ may be expressed as

N=\bigcap_{i=1}^{n}\ker\widetilde{\rho}_{i}

for some collection of irreducible representations $(\widetilde{\rho}_{i},V_{i})$ of $G$ .

Proof.

This is Problem 3.13.2 below. ∎

We conclude this section with the following result regarding characters of inflations:

Proposition 3.13.8.

Let $G$ be a finite group and $N$ a normal subgroup. For any finite-dimensional representation $(\pi,V)$ of $\widetilde{G}=G/N$ with inflation $(\widetilde{\pi},V)$ to $G$ , we have

(i)

$\chi_{\widetilde{\pi}}(g)=\chi_{\pi}(gN)$ for all $g\in G$ .
(ii)

$\|\chi_{\pi}\|_{\widetilde{G}}^{2}=\|\chi_{\widetilde{\pi}}\|^{2}_{G}$ .

Proof.

Claim (i) is more or less immediate (left as an exercise below). For (ii), use (i) and compute:

	$\displaystyle\\|\chi_{\widetilde{\pi}}\\|_{G}^{2}=$	$\displaystyle\frac{1}{\|G\|}\sum_{g\in G}\|\chi_{\widetilde{\pi}}(g)\|^{2}=\frac{1% }{\|G\|}\sum_{gN\in G/N}\sum_{n\in N}\|\chi_{\widetilde{\pi}}(gn)\|^{2}$
		$\displaystyle=\frac{1}{\|G\|}\sum_{gN\in G/N}\sum_{n\in N}\|\chi_{\pi}(gN)\|^{2}=% \frac{1}{\|G\|}\sum_{gN\in G/N}\|N\|\|\chi_{\pi}(gN)\|^{2}$
		$\displaystyle=\frac{1}{\|G\|/\|N\|}\sum_{gN\in G/N}\|\chi_{\pi}(gN)\|^{2}=\\|\chi_{% \pi}\\|^{2}_{\widetilde{G}}.$

∎

Remark 3.13.9.

Note that this proposition and Theorem 2.9.7 (ii) provide an alternative proof of Proposition 3.13.3.

3.13.2. Exercises

Problem 77. Show Proposition 3.13.8(i).

Problem 78. Prove Proposition 3.13.7 by proceeding as follows:

(a)

Given a normal subgroup $N\subseteq G$ , let $\{(\rho_{i},V_{i})\}$ denote the irreducible representations of $\widetilde{G}=G/N$ , and let $(\widetilde{\rho}_{i},V_{i})$ be the lift of $(\rho_{i},V_{i})$ to $G$ . Define

$K=\bigcap_{i=1}^{n}\ker\widetilde{\rho}_{i}.$

Show that $K$ is normal in $G$ .
(b)

Use Proposition 3.13.2 to show that $N\subseteq K$ , and hence $|G/K|\leq|G/N|$ .
(c)

Use Theorem 1.6.6 to show that $|G/N|=\sum_{i}(\operatorname{dim}V_{i})^{2}$ .
(d)

Use Propositions 3.13.2 and 3.13.3 to show that each $(\widetilde{\rho}_{i},V_{i})$ is the lift of an irreducible representation representation $(\sigma_{i},V_{i})$ of $G/K$ .
(e)

Use Theorem 1.6.6 to show that $\sum_{i}(\operatorname{dim}V_{i})^{2}\leq|G/K|$ .
(f)

Conclude that $N=K$ .

Problem 79. Using the character table of $S_{5}$ , find the character table of $A_{5}$ . Using the character table, show that $A_{5}$ is simple (that is, it has no non-trivial proper normal subgroups). Hint: every element of $A_{5}$ is conjugate to its inverse.

Problem 80. (Challenging!) Let $(\pi,V)$ be a faithful finite-dimensional representation of a finite group $G$ . Given an irreducible representation $(\rho,W)$ of $G$ , show that $(\rho,W)$ is isomorphic to a subrepresentation of $(\pi^{\otimes n},V^{\otimes n})$ for some $n\geq 1$ .

	$\displaystyle\\|\chi_{\widetilde{\pi}}\\|_{G}^{2}=$	$\displaystyle\frac{1}{\|G\|}\sum_{g\in G}\|\chi_{\widetilde{\pi}}(g)\|^{2}=\frac{1% }{\|G\|}\sum_{gN\in G/N}\sum_{n\in N}\|\chi_{\widetilde{\pi}}(gn)\|^{2}$
		$\displaystyle=\frac{1}{\|G\|}\sum_{gN\in G/N}\sum_{n\in N}\|\chi_{\pi}(gN)\|^{2}=% \frac{1}{\|G\|}\sum_{gN\in G/N}\|N\|\|\chi_{\pi}(gN)\|^{2}$
		$\displaystyle=\frac{1}{\|G\|/\|N\|}\sum_{gN\in G/N}\|\chi_{\pi}(gN)\|^{2}=\\|\chi_{% \pi}\\|^{2}_{\widetilde{G}}.$