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 allow 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:

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-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 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-theoretic

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.

Proofs

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 given by Then φ is again a projection: it is clearly K-linear, maps K[G] onto V, and induces the identity on 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 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.[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 an element of both I and V. This proves that V is not a direct complement of I for all V, so K[G] is not semisimple.

Notes

  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.

References

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.