Maschke's theorem
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In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in
group representation In the mathematical field of representation theory, group representations describe abstract groups in terms of bijective linear transformations of a vector space to itself (i.e. vector space automorphisms); in particular, they can be used to ...
theory that concerns the decomposition of representations of a finite group into
irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...
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 In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natur ...
that, while the decomposition into a direct sum of irreducible subrepresentations may not be unique, the irreducible pieces have well-defined
multiplicities In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion of multip ...
. In particular, a representation of a finite group over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
of characteristic zero is determined up to
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
by its
character Character or Characters may refer to: Arts, entertainment, and media Literature * ''Character'' (novel), a 1936 Dutch novel by Ferdinand Bordewijk * ''Characters'' (Theophrastus), a classical Greek set of character sketches attributed to The ...
.


Formulations

Maschke's theorem addresses the question: when is a general (finite-dimensional) representation built from irreducible
subrepresentation In representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algeb ...
s 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 In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
of complex-valued class functions of a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
G has a natural G-invariant
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
structure, described in the article
Schur orthogonality relations In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups. They admit a generalization to the case of compact groups in general, and in ...
. Maschke's theorem was originally proved for the case of representations over \Complex by constructing U as the
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
of W under this inner product.


Module-theoretic

One of the approaches to representations of finite groups is through
module theory In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of ''module'' generalizes also the notion of abelian group, since the abelian groups are exactly the mo ...
. ''Representations'' of a group G are replaced by ''modules'' over its group algebra K /math> (to be precise, there is an
isomorphism of categories In category theory, two categories ''C'' and ''D'' are isomorphic if there exist functors ''F'' : ''C'' → ''D'' and ''G'' : ''D'' → ''C'' which are mutually inverse to each other, i.e. ''FG'' = 1''D'' (the identity functor on ''D'') and ''GF' ...
between K text and \operatorname_, the
category of representations In representation theory, the category of representations of some algebraic structure has the representations of as objects and equivariant maps as morphisms between them. One of the basic thrusts of representation theory is to understand the con ...
of G). Irreducible representations correspond to
simple module In mathematics, specifically in ring theory, the simple modules over a ring ''R'' are the (left or right) modules over ''R'' that are non-zero and have no non-zero proper submodules. Equivalently, a module ''M'' is simple if and only if every cy ...
s. In the module-theoretic language, Maschke's theorem asks: is an arbitrary module
semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
? 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 In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . Examples As an example, the field of real numbers is not algebraically closed, because ...
, for example, K is a field of
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
or
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
numbers, then a somewhat more complicated statement holds: the group algebra K /math> is a product of matrix algebras over
division ring In algebra, a division ring, also called a skew field, is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicative inverse, that is, an element ...
s 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 In mathematics, and more specifically in homological algebra, the splitting lemma states that in any abelian category, the following statements are equivalent for a short exact sequence : 0 \longrightarrow A \mathrel B \mathrel C \longrightarro ...
, 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 mathematics, the concept of an inverse element generalises the concepts of opposite () and reciprocal () of numbers. Given an operation denoted here , and an identity element denoted , if , one says that is a left inverse of , and that is ...
in ''K''. This might lead one to ask if the
converse Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a definite or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical c ...
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 In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
for \begin 1 & 1 \\ 0 & 1 \end, and the only eigenvector for that is \begin 1 \\ 0 \end. * Consider a
prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
''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

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