Maschke's theorem explained

In mathematics, Maschke's theorem,^{[1] [2]} 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

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[G] (to be precise, there is an isomorphism of categories between K[G] -Mod and \operatorname{Rep} _{G} , 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, their classification as given by the Wedderburn–Artin theorem. When

K is the field of complex numbers, this shows that the algebra K[G] is a product of several copies of complex matrix algebras, one for each irreducible representation.^[3] If the field K has characteristic zero, but is not algebraically closed, for example if K is the field of real or rational numbers, then a somewhat more complicated statement holds: the group algebra K[G] is a product of matrix algebras over division rings over K . The summands correspond to irreducible representations of G over K .^[4]

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 $p(x)\; =\; \backslash frac\; \backslash sum\_\; g\; \backslash cdot\; p\_0\; \backslash cdot\; g^\; (x)$, where

g ⋅ p _{0} ⋅ g ^{-1} is an abbreviation of \rho _{W{g}} ⋅ p _{0} ⋅ \rho ^{-1} _{V{g} }, with \rho _{W{g},} \rho ^{-1} _{V{g} } 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 ,

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 = \{ 0 \} , 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 ⊕ \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 $$\backslash begin\backslash varphi:K[G]\backslash to\; V\; \backslash \backslash \backslash varphi:x\; \backslash mapsto\; \backslash frac\backslash sum\_\; s\backslash cdot\; \backslash pi(s^\; \backslash cdot\; x)\backslash end$$

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

so φ is in fact K[''G'']-linear. By the splitting lemma,

K[G] = V ⊕ \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 $x\; =\; \backslash sum\backslash lambda\_g\; g\backslash in\; K[G]$ define $\backslash epsilon(x)\; =\; \backslash sum\backslash lambda\_g$. 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 ≠ 0 . Let V be given, and let $v=\backslash sum\backslash mu\_gg$ be any nonzero element of V. If \epsilon(v) =0 , the claim is immediate. Otherwise, let $s\; =\; \backslash sum\; 1\; g$. Then \epsilon(s) = \#G ⋅ 1 = 0 so s \in I and $$sv\; =\; \backslash left(\backslash sum1g\backslash right)\backslash !\backslash left(\backslash sum\backslash mu\_gg\backslash right)\; =\; \backslash sum\backslash epsilon(v)g\; =\; \backslash epsilon(v)s$$

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

Z and the representation \rho: Z \to GL _{2(\Complex)} defined by \rho(n) = \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} ^{n} = \begin{bmatrix} 1 & n \ 0 & 1 \end{bmatrix} . Let W = \Complex ⋅ \begin{bmatrix} 1 \ 0 \end{bmatrix} , a 1-dimensional subspace of \Complex ^{2} spanned by \begin{bmatrix} 1 \ 0 \end{bmatrix} . Then the restriction of \rho on W is a trivial subrepresentation of Z . However, there's no U such that both W, U are subrepresentations of Z and \Complex ^{2} = W ⊕ U : any such U needs to be 1-dimensional, but any 1-dimensional subspace preserved by \rho has to be spanned by an eigenvector for \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} , and the only eigenvector for that is \begin{bmatrix} 1 \ 0 \end{bmatrix} .

• Consider a prime p, and the group

Z /p Z , field K = F _{p} , and the representation \rho: Z /p Z \to GL _{2(F} _{p)} defined by \rho(n) = \begin{bmatrix} 1 & n \ 0 & 1 \end{bmatrix} . Simple calculations show that there is only one eigenvector for \begin{bmatrix} 1 & 1 \ 0 & 1 \end{bmatrix} here, so by the same argument, the 1-dimensional subrepresentation of Z /p Z is unique, and Z /p Z cannot be decomposed into the direct sum of two 1-dimensional subrepresentations.

Notes

1. Maschke . Heinrich . 1898-07-22 . Ueber den arithmetischen Charakter der Coefficienten der Substitutionen endlicher linearer Substitutionsgruppen . German . On the arithmetical character of the coefficients of the substitutions of finite linear substitution groups . Math. Ann. . 50 . 4 . 492–498 . 29.0114.03 . 1511011 . 10.1007/BF01444297 .
2. Maschke . Heinrich . 1899-07-27 . Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einige durchgehends verschwindende Coefficienten auftreten, intransitiv sind . German . Proof of the theorem that those finite linear substitution groups, in which some everywhere vanishing coefficients appear, are intransitive . Math. Ann. . 52 . 2–3 . 363–368 . 30.0131.01 . 1511061 . 10.1007/BF01476165 .
3. The number of the summands can be computed, and turns out to be equal to the number of the conjugacy classes of the group.
4. 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.

References

• Book: Lang , Serge . Serge Lang

. Serge Lang . Algebra . Revised 3rd . Graduate Texts in Mathematics, 211 . . New York . 2002-01-08 . 978-0-387-95385-4 . 1878556 . 0984.00001.

• Book: Serre , Jean-Pierre . Jean-Pierre Serre

. Jean-Pierre Serre . Linear Representations of Finite Groups . registration . Graduate Texts in Mathematics, 42 . . New York–Heidelberg . 1977-09-01 . 978-0-387-90190-9 . 0450380 . 0355.20006.