Maschke's theorem

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.

Contents 1 Formulations 1.1 Group-theoretic 1.2 Module-theoretic 1.3 Category-theoretic 2 Proofs 2.1 Group-theoretic 2.2 Module-theoretic 3 Converse statement 4 Non-examples 5 Notes 6 References 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 —  {displaystyle V} is a representation of a finite group {displaystyle G} over a field {displaystyle mathbb {F} } with characteristic not dividing the order of {displaystyle G} . If {displaystyle V} has a subrepresentation {displaystyle W} , then it has another subrepresentation {displaystyle U} such that {displaystyle V=Woplus U} .[4][5] Then the corollary is Corollary (Maschke's theorem) — Every representation of a finite group {displaystyle G} over a field {displaystyle mathbb {F} } with characteristic not dividing the order of {displaystyle G} is a direct sum of irreducible representations.[6][7] The vector space of complex-valued class functions of a group {displaystyle G} has a natural {displaystyle G} -invariant inner product structure, described in the article Schur orthogonality relations. Maschke's theorem was originally proved for the case of representations over {displaystyle mathbb {C} } by constructing {displaystyle U} as the orthogonal complement of {displaystyle W} under this inner product.

Module-theoretic One of the approaches to representations of finite groups is through module theory. Representations of a group {displaystyle G} are replaced by modules over its group algebra  {displaystyle K[G]} (to be precise, there is an isomorphism of categories between {displaystyle K[G]{text{-Mod}}} and {displaystyle operatorname {Rep} _{G}} , the category of representations of {displaystyle 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 {displaystyle G} be a finite group and {displaystyle K} a field whose characteristic does not divide the order of {displaystyle G} . Then {displaystyle K[G]} , the group algebra of {displaystyle 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 {displaystyle K} is the field of complex numbers, this shows that the algebra {displaystyle K[G]} is a product of several copies of complex matrix algebras, one for each irreducible representation.[10] If the field {displaystyle K} has characteristic zero, but is not algebraically closed, for example, {displaystyle K} is a field of real or rational numbers, then a somewhat more complicated statement holds: the group algebra {displaystyle K[G]} is a product of matrix algebras over division rings over {displaystyle K} . The summands correspond to irreducible representations of {displaystyle G} over {displaystyle 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 Group-theoretic Let U be a subspace of V complement of W. Let {displaystyle p_{0}:Vto W} be the projection function, i.e., {displaystyle p_{0}(w+u)=w} for any {displaystyle uin U,win W} .

Define {textstyle p(x)={frac {1}{#G}}sum _{gin G}gcdot p_{0}cdot g^{-1}(x)} , where {displaystyle gcdot p_{0}cdot g^{-1}} is an abbreviation of {displaystyle rho _{W}{g}cdot p_{0}cdot rho _{V}{g^{-1}}} , with {displaystyle rho _{W}{g},rho _{V}{g^{-1}}} being the representation of G on W and V. Then, {displaystyle ker p} is preserved by G under representation {displaystyle rho _{V}} : for any {displaystyle w'in ker p,hin G} , {displaystyle {begin{aligned}p(hw')&=hcdot h^{-1}{frac {1}{#G}}sum _{gin G}gcdot p_{0}cdot g^{-1}(hw')\&=hcdot {frac {1}{#G}}sum _{gin G}(h^{-1}cdot g)cdot p_{0}cdot (g^{-1}h)w'\&=hcdot {frac {1}{#G}}sum _{gin G}gcdot p_{0}cdot g^{-1}w'\&=hcdot p(w')\&=0end{aligned}}} so {displaystyle w'in ker p} implies that {displaystyle hw'in ker p} . So the restriction of {displaystyle rho _{V}} on {displaystyle ker p} is also a representation.

By the definition of {displaystyle p} , for any {displaystyle win W} , {displaystyle p(w)=w} , so {displaystyle Wcap ker p={0}} , and for any {displaystyle vin V} , {displaystyle p(p(v))=p(v)} . Thus, {displaystyle p(v-p(v))=0} , and {displaystyle v-p(v)in ker p} . Therefore, {displaystyle V=Woplus 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 {displaystyle {begin{cases}varphi :K[G]to V\varphi :xmapsto {frac {1}{#G}}sum _{sin G}scdot pi (s^{-1}cdot x)end{cases}}} 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 {displaystyle {begin{aligned}varphi (tcdot x)&={frac {1}{#G}}sum _{sin G}scdot pi (s^{-1}cdot tcdot x)\&={frac {1}{#G}}sum _{uin G}tcdot ucdot pi (u^{-1}cdot x)\&=tcdot varphi (x),end{aligned}}} so φ is in fact K[G]-linear. By the splitting lemma, {displaystyle K[G]=Voplus 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.[12] Proof. For {textstyle x=sum lambda _{g}gin K[G]} define {textstyle epsilon (x)=sum lambda _{g}} . Let {displaystyle I=ker epsilon } . Then I is a K[G]-submodule. We will prove that for every nontrivial submodule V of K[G], {displaystyle Icap Vneq 0} . Let V be given, and let {textstyle v=sum mu _{g}g} be any nonzero element of V. If {displaystyle epsilon (v)=0} , the claim is immediate. Otherwise, let {textstyle s=sum 1g} . Then {displaystyle epsilon (s)=#Gcdot 1=0} so {displaystyle sin I} and {displaystyle sv=left(sum 1gright)!left(sum mu _{g}gright)=sum epsilon (v)g=epsilon (v)s} so that {displaystyle 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 {displaystyle mathbb {Z} } and the representation {displaystyle rho :mathbb {Z} to mathrm {GL} _{2}(mathbb {C} )} defined by {displaystyle rho (n)={begin{bmatrix}1&1\0&1end{bmatrix}}^{n}={begin{bmatrix}1&n\0&1end{bmatrix}}} . Let {displaystyle W=mathbb {C} cdot {begin{bmatrix}1\0end{bmatrix}}} , a 1-dimensional subspace of {displaystyle mathrm {GL} _{2}(mathbb {C} )} spanned by {displaystyle {begin{bmatrix}1\0end{bmatrix}}} . Then the restriction of {displaystyle rho } on W is a trivial subrepresentation of {displaystyle mathbb {Z} } . However, there's no U such that both W, U are subrepresentations of {displaystyle mathbb {Z} } and {displaystyle mathbb {Z} =Woplus U} : any such U needs to be 1-dimensional, but any 1-dimensional subspace preserved by {displaystyle rho } has to be spanned by eigenvector for {displaystyle {begin{bmatrix}1&1\0&1end{bmatrix}}} , and the only eigenvector for that is {displaystyle {begin{bmatrix}1\0end{bmatrix}}} . Consider a prime p, and the group {displaystyle mathbb {Z} /pmathbb {Z} } , field {displaystyle K=mathbb {F} _{p}} , and the representation {displaystyle rho :mathbb {Z} /pmathbb {Z} to mathrm {GL} _{2}(mathbb {F} _{p})} defined by {displaystyle rho (n)={begin{bmatrix}1&n\0&1end{bmatrix}}} . Simple calculations show that there is only one eigenvector for {displaystyle {begin{bmatrix}1&1\0&1end{bmatrix}}} here, so by the same argument, the 1-dimensional subrepresentation of {displaystyle mathbb {Z} /pmathbb {Z} } is unique, and {displaystyle mathbb {Z} /pmathbb {Z} } cannot be decomposed into the direct sum of two 1-dimensional subrepresentations. Notes ^ 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. ^ 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. ^ O'Connor, John J.; Robertson, Edmund F., "Heinrich Maschke", MacTutor History of Mathematics archive, University of St Andrews ^ Fulton & Harris, Proposition 1.5. ^ Serre, Theorem 1. ^ Fulton & Harris, Corollary 1.6. ^ Serre, Theorem 2. ^ It follows that every module over {displaystyle K[G]} is a semisimple module. ^ 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. ^ The number of the summands can be computed, and turns out to be equal to the number of the conjugacy classes of the group. ^ 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. ^ Serre, Exercise 6.1. References Lang, Serge (2002-01-08). Algebra. Graduate Texts in Mathematics, 211 (Revised 3rd ed.). New York: Springer-Verlag. ISBN 978-0-387-95385-4. MR 1878556. Zbl 0984.00001. Serre, Jean-Pierre (1977-09-01). Linear Representations of Finite Groups. Graduate Texts in Mathematics, 42. New York–Heidelberg: Springer-Verlag. ISBN 978-0-387-90190-9. MR 0450380. Zbl 0355.20006. Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts in Mathematics, Readings in Mathematics. Vol. 129. New York: Springer-Verlag. doi:10.1007/978-1-4612-0979-9. ISBN 978-0-387-97495-8. MR 1153249. OCLC 246650103. Categories: Representation theory of finite groupsTheorems in group theoryTheorems in representation theory