Maschke's theorem

In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make general conclusions about representations of a finite group ''G'' without actually computing them. It reduces the task of classifying all representations to a more manageable task of classifying irreducible representations, since when the theorem applies, any representation is a direct sum of irreducible pieces (constituents). Moreover, it follows from the Jordan–Hölder theorem that, while the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined multiplicities. In particular, a representation of a finite group over a field of characteristic zero is determined up to isomorphism by its character.

Formulations

Maschke's theorem addresses the question: when is a general (finite-dimensional) representation built from irreducible subrepresentations using the direct sum operation? This question (and its answer) are formulated differently for different perspectives on group representation theory.

Group-theoretic

Maschke's theorem is commonly formulated as a corollary to the following result:

Then the corollary is

The vector space of complex-valued class functions of a group $G$ has a natural $G$-invariant inner product structure, described in the article Schur orthogonality relations. Maschke's theorem was originally proved for the case of representations over $\Complex$ by constructing $U$ as the orthogonal complement of $W$ under this inner product.

Module-theoretic

One of the approaches to representations of finite groups is through module theory. ''Representations'' of a group $G$ are replaced by ''modules'' over its group algebra K /math> (to be precise, there is an isomorphism of categories between K text and $\operatorname_$, the category of representations of $G$). Irreducible representations correspond to simple modules. In the module-theoretic language, Maschke's theorem asks: is an arbitrary module semisimple? In this context, the theorem can be reformulated as follows:

The importance of this result stems from the well developed theory of semisimple rings, in particular, the Artin–Wedderburn theorem (sometimes referred to as Wedderburn's Structure Theorem). When $K$ is the field of complex numbers, this shows that the algebra K /math> is a product of several copies of complex matrix algebras, one for each irreducible representation. If the field $K$ has characteristic zero, but is not algebraically closed, for example, $K$ is a field of real or rational numbers, then a somewhat more complicated statement holds: the group algebra K /math> is a product of matrix algebras over division rings over $K$. The summands correspond to irreducible representations of $G$ over $K$.One must be careful, since a representation may decompose differently over different fields: a representation may be irreducible over the real numbers but not over the complex numbers.

Category-theoretic

Reformulated in the language of semi-simple categories, Maschke's theorem states

Proofs

Group-theoretic

Let ''U'' be a subspace of ''V'' complement of ''W''. Let $p_0 : V \to W$ be the projection function, i.e., $p_0(w + u) = w$ for any $u \in U, w \in W$.

Define $p(x) = \frac \sum_ g \cdot p_0 \cdot g^ (x)$, where $g \cdot p_0 \cdot g^$ is an abbreviation of $\rho_W \cdot p_0 \cdot \rho_V$, with $\rho_W, \rho_V$ being the representation of ''G'' on ''W and'' ''V''. Then, $\ker p$ is preserved by ''G'' under representation $\rho_V$: for any $w' \in \ker p, h \in G$,
$\begin p(hw') &= h \cdot h^ \frac \sum_ g \cdot p_0 \cdot g^ (hw') \\ &= h \cdot \frac \sum_ (h^ \cdot g) \cdot p_0 \cdot (g^ h) w' \\ &= h \cdot \frac \sum_ g \cdot p_0 \cdot g^ w' \\ &= h \cdot p(w') \\ &= 0 \end$

so $w' \in \ker p$ implies that $hw' \in \ker p$. So the restriction of $\rho_V$ on $\ker p$ is also a representation.

By the definition of $p$, for any $w \in W$, $p(w) = w$, so $W \cap \ker\ p = \$, and for any $v \in V$, $p(p(v)) = p(v)$. Thus, $p(v-p(v)) = 0$, and $v - p(v) \in \ker p$. Therefore, $V = W \oplus \ker p$.

Module-theoretic

Let ''V'' be a ''K'' 'G''submodule. We will prove that ''V'' is a direct summand. Let ''π'' be any ''K''-linear projection of ''K'' 'G''onto ''V''. Consider the map
\begin
\varphi:K to V \\
\varphi:x \mapsto \frac\sum_ s\cdot \pi(s^ \cdot x)
\end

Then ''φ'' is again a projection: it is clearly ''K''-linear, maps ''K'' 'G''to ''V'', and induces the identity on ''V'' (therefore, maps ''K'' 'G''onto ''V''). Moreover we have

$\begin \varphi(t\cdot x) &= \frac\sum_ s\cdot \pi(s^\cdot t\cdot x)\\ &= \frac\sum_ t\cdot u\cdot \pi(u^\cdot x)\\ &= t\cdot\varphi(x), \end$

so ''φ'' is in fact ''K'' 'G''linear. By the splitting lemma, K V \oplus \ker \varphi. This proves that every submodule is a direct summand, that is, ''K'' 'G''is semisimple.

Converse statement

The above proof depends on the fact that #''G'' is invertible in ''K''. This might lead one to ask if the converse of Maschke's theorem also holds: if the characteristic of ''K'' divides the order of ''G'', does it follow that ''K'' 'G''is not semisimple? The answer is ''yes''.

Proof. For x = \sum\lambda_g g\in K /math> define $\epsilon(x) = \sum\lambda_g$. Let $I=\ker\epsilon$. Then ''I'' is a ''K'' 'G''submodule. We will prove that for every nontrivial submodule ''V'' of ''K'' 'G'' $I \cap V \neq 0$. Let ''V'' be given, and let $v=\sum\mu_gg$ be any nonzero element of ''V''. If $\epsilon(v)=0$, the claim is immediate. Otherwise, let $s = \sum 1 g$. Then $\epsilon(s) = \#G \cdot 1 = 0$ so $s \in I$ and
$sv = \left(\sum1g\right)\!\left(\sum\mu_gg\right) = \sum\epsilon(v)g = \epsilon(v)s$

so that $sv$ is a nonzero element of both ''I'' and ''V''. This proves ''V'' is not a direct complement of ''I'' for all ''V'', so ''K'' 'G''is not semisimple.

Non-examples

The theorem can not apply to the case where ''G'' is infinite, or when the field ''K'' has characteristics dividing #''G''. For example,

* Consider the infinite group $\mathbb$ and the representation $\rho: \mathbb \to \mathrm_2(\Complex)$ defined by $\rho(n) = \begin 1 & 1 \\ 0 & 1 \end^n = \begin 1 & n \\ 0 & 1 \end$. Let $W = \Complex \cdot \begin 1 \\ 0 \end$, a 1-dimensional subspace of $\mathrm_2(\Complex)$ spanned by $\begin 1 \\ 0 \end$. Then the restriction of $\rho$ on ''W'' is a trivial subrepresentation of $\mathbb$. However, there's no ''U'' such that both ''W, U'' are subrepresentations of $\mathbb$ and $\mathbb = W \oplus U$: any such ''U'' needs to be 1-dimensional, but any 1-dimensional subspace preserved by $\rho$ has to be spanned by eigenvector for $\begin 1 & 1 \\ 0 & 1 \end$, and the only eigenvector for that is $\begin 1 \\ 0 \end$.
* Consider a prime ''p'', and the group $\mathbb/p\mathbb$, field $K = \mathbb_p$, and the representation $\rho: \mathbb/p\mathbb \to \mathrm_2(\mathbb_p)$ defined by $\rho(n) = \begin 1 & n \\ 0 & 1 \end$. Simple calculations show that there is only one eigenvector for $\begin 1 & 1 \\ 0 & 1 \end$ here, so by the same argument, the 1-dimensional subrepresentation of $\mathbb/p\mathbb$ is unique, and $\mathbb/p\mathbb$ cannot be decomposed into the direct sum of two 1-dimensional subrepresentations.

Notes

References

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{{DEFAULTSORT:Maschke's Theorem
Representation theory of finite groups
Theorems in group theory
Theorems in representation theory