Artin's theorem on induced characters

In group representation theory, a branch of mathematics, Artin's theorem on induced characters, introduced by E. Artin, states that a character on a finite group is a rational linear combination of characters induced from all cyclic subgroups of the group.

There is a similar but in some sense more precise theorem due to Brauer, which says that the theorem remains true if "rational" and "cyclic subgroup" are replaced with "integer" and "elementary subgroup".

In Linear Representation of Finite Groups Serre states in Chapter 9.2, 17 ^[1] the theorem in the following, more general way:

Let ${\displaystyle G}$ be a finite group and ${\displaystyle X}$ any family of subgroups.

Then the following are equivalent:

1. ${\displaystyle G}$ is the union of conjugates of the subgroups in ${\displaystyle X}$: ${\displaystyle G=\bigcup _{g\in G,H\in X}g^{-1}Hg}$
2. for every character ${\displaystyle \chi }$ of ${\displaystyle G}$ there exist characters ${\displaystyle \chi _{H}}$ of ${\displaystyle H}$ for each ${\displaystyle H\in X}$ and ${\displaystyle n\in \mathbb {N} }$ such that ${\displaystyle n\chi =\sum _{H\in X}{\text{Ind}}_{H}^{G}(\chi _{H})}$

This in turn implies Artin's original statement, by choosing ${\displaystyle X}$ to be the set of all cyclic subgroups of ${\displaystyle G}$.

Let ${\displaystyle G}$ be a finite group and ${\displaystyle (\chi _{i})}$ its irreducible characters. Recall the representation ring ${\displaystyle {\mathcal {R}}(G)}$ is the free abelian group on the set ${\displaystyle \{\chi _{i}\}}$. Since all of ${\displaystyle G}$'s characters are linear combinations of ${\displaystyle (\chi _{i})}$ with nonnegative integer coefficients, every element of ${\displaystyle {\mathcal {R}}(G)}$ is the difference of two characters of ${\displaystyle G}$. Moreover, because the product of two characters is also a character, ${\displaystyle {\mathcal {R}}(G)}$ is a ring. It is a sub-ring of the ${\displaystyle \mathbb {C} }$-algebra of class functions on ${\displaystyle G}$. This algebra is isomorphic to ${\displaystyle \mathbb {C} \otimes {\mathcal {R}}(G)}$, and has ${\displaystyle (\chi _{i})}$ as a basis.

Both the operation ${\displaystyle {\text{Res}}}$ of restricting a representation of ${\displaystyle G}$ to one of its subgroups ${\displaystyle H}$ and the adjoint operation ${\displaystyle {\text{Ind}}}$ of inducing representations from ${\displaystyle H}$ to ${\displaystyle G}$ give abelian group homomorphisms:

${\displaystyle {\begin{array}{cccl}{\text{Res}}\colon &{\mathcal {R}}(G)&\to &{\mathcal {R}}(H)\\&\sum k_{i}\chi _{i}&\mapsto &\sum k_{i}{\text{Res}}_{H}^{G}\chi _{i}\end{array}}}$

${\displaystyle {\begin{array}{cccl}{\text{Ind}}\colon &{\mathcal {R}}(H)&\to &\,{\mathcal {R}}(G)\\&\sum k_{i}\chi _{i}&\mapsto &\sum k_{i}{\text{Ind}}_{H}^{G}\chi _{i}\end{array}}}$

where ${\displaystyle {\text{Res}}}$ is actually a ring homomorphism.

With these notations, the theorem can be equivalently rewritten as follows. If ${\displaystyle X}$ is a family of subgroups of ${\displaystyle G}$, the following properties are equivalent:

1. ${\displaystyle G}$ is the union of the conjugates of the subgroups in ${\displaystyle X}$
2. The cokernel of ${\displaystyle {\text{Ind}}\colon \bigoplus _{H\in X}{\mathcal {R}}(H)\to {\mathcal {R}}(G)}$ is finite.

We start with the following lemma:

Lemma. Let ${\displaystyle H}$ be an element of ${\displaystyle X}$. We claim that for every ${\displaystyle f\in {\mathcal {R}}(H)}$, ${\displaystyle {\text{Ind}}(f)}$ vanishes on ${\displaystyle g\in G}$ if ${\displaystyle g}$ is not conjugate to any ${\displaystyle h\in H}$.

Proof. It is enough to prove this lemma for the character ${\displaystyle \phi }$ of a representation ${\displaystyle \theta \colon H\to \mathrm {GL} (W)}$, since any ${\displaystyle f\in {\mathcal {R}}(H)}$ is a difference of two such characters. So, let ${\displaystyle \rho \colon G\to \mathrm {GL} (V)}$ be the representation of ${\displaystyle G}$ induced from the representation ${\displaystyle \theta }$ of ${\displaystyle H}$, and let ${\displaystyle (r_{i})}$ be a set of representatives of the cosets of ${\displaystyle H}$ in ${\displaystyle G}$, which are the points of ${\displaystyle G/H}$. By definition of induced representation, ${\displaystyle V}$ is the direct sum of its subspaces ${\displaystyle \rho (r_{i})W}$, and the linear transformation ${\displaystyle \rho (g)}$ permutes these subspaces, since

${\displaystyle \rho (g)\circ \rho (r_{i})W=\rho (gr_{i})W=\rho (r_{i'})W}$

where ${\displaystyle gr_{i}=r_{i'}h}$ for some ${\displaystyle h\in H}$. To show that ${\displaystyle {\text{Ind}}(\phi )(g)={\text{tr}}_{V}(\rho (g))}$ vanishes, we now choose a basis for ${\displaystyle V}$ that is a union of the bases of the subspaces ${\displaystyle \rho (r_{i})W}$. In this basis for ${\displaystyle V}$, the diagonal matrix entry of ${\displaystyle \rho (g)}$ vanishes for each basis vector in ${\displaystyle \rho (r_{i})W}$ with ${\displaystyle r_{i}\neq r_{i'}}$. But ${\displaystyle r_{i}=r_{i'}}$ would imply ${\displaystyle r_{i}^{-1}gr_{i}=h\in H}$, which is ruled out by our assumption that ${\displaystyle g}$ is not conjugate to any element of ${\displaystyle H}$. Thus, all the diagonal matrix entries of ${\displaystyle \rho (g)}$ vanish, so ${\displaystyle {\text{tr}}_{V}(\rho _{g})=0}$ as desired, proving the lemma. ■

Now we prove 2. ${\displaystyle \implies }$ 1. The lemma implies that all elements in the image of ${\displaystyle {\text{Ind}}\colon \bigoplus _{H\in X}{\mathcal {R}}(H)\to {\mathcal {R}}(G)}$ vanish on every ${\displaystyle g\in G}$ in

${\displaystyle S:=G-\bigcup _{g\in G,H\in X}g^{-1}Hg}$,

The same therefore holds for all elements in the image of the ${\displaystyle \mathbb {C} }$-linear map

${\displaystyle \mathbb {C} \otimes {\text{Ind}}\colon \mathbb {C} \otimes \bigoplus _{H\in X}{\mathcal {R}}(H)\to \mathbb {C} \otimes {\mathcal {R}}(G)}$

On the other hand, this map is surjective, because otherwise ${\displaystyle {\text{Ind}}}$ would have an infinite cokernel, contradicting assumption 2. Thus, every element of ${\displaystyle \mathbb {C} \otimes {\mathcal {R}}(G)}$ vanishes on ${\displaystyle S}$, insuring ${\displaystyle S=\emptyset }$, so that every element of ${\displaystyle G}$ is conjugate to an element of some subgroup ${\displaystyle H\in X}$, as was to be shown.

Let us now prove 1. ${\displaystyle \implies }$ 2. First, note that it is enough to show 1. implies that ${\displaystyle \mathbb {C} \otimes {\text{Ind}}\colon \mathbb {C} \otimes \bigoplus _{H\in X}{\mathcal {R}}(H)\to \mathbb {C} \otimes {\mathcal {R}}(G)}$ is surjective. Indeed, this surjectivity implies that ${\displaystyle \mathbb {C} \otimes {\mathcal {R}}(G)}$ has a basis ${\displaystyle (e_{i})}$ composed of elements of the image ${\displaystyle A}$ of ${\displaystyle {\text{Ind}}}$. Since this basis must have the same cardinality ${\displaystyle n}$ as ${\displaystyle (\chi _{i})}$, the quotient ${\displaystyle {\mathcal {R}}(G)/A}$ is isomorphic to some quotient ${\displaystyle \mathbb {Z} ^{n}/\prod _{i}^{n}H_{i}\cong \prod _{i}^{n}\mathbb {Z} /H_{i}}$ where the ${\displaystyle H_{i}}$ are non-trivial ideals of ${\displaystyle \mathbb {Z} }$, and this quotient is clearly finite, giving 2.

By duality, proving the surjectivity of ${\displaystyle \mathbb {C} \otimes {\text{Ind}}}$ is equivalent to proving the injectivity of

${\displaystyle \mathbb {C} \otimes {\text{Res}}\colon \mathbb {C} \otimes {\mathcal {R}}(G)\to \mathbb {C} \otimes \bigoplus _{H\in X}{\mathcal {R}}(H).}$

However, this is clearly true, because it is equivalent to saying that if a character vanishes on every conjugacy class of ${\displaystyle G}$ it vanishes, which holds because characters are constant on each conjugacy class.

This concludes the proof of the theorem.

• R. Brauer and J. Tate. On the characters of finite groups. Ann. of Math., 6.2 (1955), p. 1-7.
• https://www.math.toronto.edu/murnaghan/courses/mat445/artinbrauer.pdf
• https://math.stackexchange.com/questions/4854649/artins-theorem-for-the-linear-representation-of-finite-groups/4854696#4854696