Maschke's theorem

In mathematics, Maschke's theorem,[1][2] named after Heinrich Maschke,[3] 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.

FormulationsEdit

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-theoreticEdit

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

Theorem. If V is a complex representation of a finite group G with a subrepresentation W, then there is another subrepresentation U of V such that V=WU.[4][5]

Then the corollary is

Corollary (Maschke's theorem). Every representation of a finite group G over a field F with characteristic not dividing the order of G is a direct sum of irreducible representations.[6][7]

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   by constructing U as the orthogonal complement of W under this inner product.

Module-theoreticEdit

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 RepG, 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:

Maschke's Theorem. Let G be a finite group and K a field whose characteristic does not divide the order of G. Then K[G], the group algebra of G, is semisimple.[8][9]

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[G] is a product of several copies of complex matrix algebras, one for each irreducible representation.[10] 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[G] is a product of matrix algebras over division rings over K. The summands correspond to irreducible representations of G over K.[11]

Category-theoreticEdit

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

Maschke's theorem. If G is a group and F is a field with characteristic not dividing the order of G, then the category of representations of G over F is semi-simple.

ProofsEdit

Group-theoreticEdit

Let U be a subspace of V complement of W. Let   be the projection function, i.e.,   for any  .

Define  , where   is an abbreviation of  , with   being the representation of G on W and V. Then,   is preserved by G under representation  : for any  ,

 

so   implies that  . So the restriction of   on   is also a representation.

By the definition of  , for any  ,  , so  , and for any  ,  . Thus,  , and  . Therefore,  .

Module-theoreticEdit

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

 

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,  . This proves that every submodule is a direct summand, that is, K[G] is semisimple.

Converse statementEdit

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.[12]

Proof. For   define  . Let  . Then I is a K[G]-submodule. We will prove that for every nontrivial submodule V of K[G],  . Let V be given, and let   be any nonzero element of V. If  , the claim is immediate. Otherwise, let  . Then   so   and

 

so that   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-examplesEdit

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   and the representation   defined by  . Let  , a 1-dimensional subspace of   spanned by  . Then the restriction of   on W is a trivial subrepresentation of  . However, there's no U such that both W, U are subrepresentations of   and  : any such U needs to be 1-dimensional, but any 1-dimensional subspace preserved by   has to be spanned by eigenvector for  , and the only eigenvector for that is  .
  • Consider a prime p, and the group  , field  , and the representation   defined by  . Simple calculations show that there is only one eigenvector for   here, so by the same argument, the 1-dim subrepresentation of   is unique, and   cannot be decomposed into the direct sum of two 1-dimensional subrepresentations.

NotesEdit

  1. ^ Maschke, Heinrich (1898-07-22). "Ueber den arithmetischen Charakter der Coefficienten der Substitutionen endlicher linearer Substitutionsgruppen" [On the arithmetical character of the coefficients of the substitutions of finite linear substitution groups]. Math. Ann. (in German). 50 (4): 492–498. doi:10.1007/BF01444297. JFM 29.0114.03. MR 1511011.
  2. ^ Maschke, Heinrich (1899-07-27). "Beweis des Satzes, dass diejenigen endlichen linearen Substitutionsgruppen, in welchen einige durchgehends verschwindende Coefficienten auftreten, intransitiv sind" [Proof of the theorem that those finite linear substitution groups, in which some everywhere vanishing coefficients appear, are intransitive]. Math. Ann. (in German). 52 (2–3): 363–368. doi:10.1007/BF01476165. JFM 30.0131.01. MR 1511061.
  3. ^ O'Connor, John J.; Robertson, Edmund F., "Heinrich Maschke", MacTutor History of Mathematics archive, University of St Andrews.
  4. ^ Fulton & Harris, Proposition 1.5.
  5. ^ Serre, Theorem 1.
  6. ^ Fulton & Harris, Corollary 1.6.
  7. ^ Serre, Theorem 2.
  8. ^ It follows that every module over K[G] is a semisimple module.
  9. ^ The converse statement also holds: if the characteristic of the field divides the order of the group (the modular case), then the group algebra is not semisimple.
  10. ^ The number of the summands can be computed, and turns out to be equal to the number of the conjugacy classes of the group.
  11. ^ 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.
  12. ^ Serre, Exercise 6.1.

ReferencesEdit