By Gregory Karpilovsky
Permit N be a regular subgroup of a finite team G and allow F be a box. a massive technique for developing irreducible FG-modules contains the appliance (perhaps repeated) of 3 uncomplicated operations: (i) restrict to FN. (ii) extension from FN. (iii) induction from FN. this is often the `Clifford idea' built by way of Clifford in 1937. some time past 20 years, the idea has loved a interval of lively improvement. the rules were reinforced and reorganized from new issues of view, particularly from the perspective of graded earrings and crossed items. the aim of this monograph is to tie jointly a number of threads of the advance in an effort to provide a finished photo of the present kingdom of the topic. it truly is assumed that the reader has had the an identical of a typical first-year graduate algebra path, i.e. familiarity with uncomplicated ring-theoretic, number-theoretic and group-theoretic techniques, and an realizing of ordinary houses of modules, tensor items and fields.
Read or Download Clifford Theory for Group Representations PDF
Similar women authors books
First released in 1920, this variation is attribute of Teasdale's paintings - brief poems of large emotional and lyrical grace, and entire of premonitions of loss of life, loss and grief, and loving appreciation of the wildlife.
Works of technological know-how fiction and delusion more and more discover gender matters, function ladies as vital characters, and are written by way of ladies writers. This publication examines women's contributions to technological know-how fiction and delusion throughout a number media and genres, reminiscent of fiction, nonfiction, movie, tv, artwork, comics, image novels, and track.
Ladies Latin Poets addresses women's dating to tradition among the 1st century B. C. and the eighteenth century A. D. through learning women's poetry in Latin. established fullyyt on unique archival learn in twelve nations, Stevenson recovers a facet of historical past usually deemed to not exist: girls who accomplished public popularity of their personal time, occasionally to a startling volume.
Extra info for Clifford Theory for Group Representations
I) An R-module (ii) Every maximal left ideal of R contains a primitive ideal and every primitive ideal is the intersection of the maximal left ideals containing it. (iii) J ( R ) is the intersection of the annihilators of irreducible R- modules. I n particular, J(R) i s a n ideal of R . Proof. (i) Clearly R/X is irreducible. Conversely, assume that V is irreducible and let v be a nonzero element of V. Then Rv is a nonzero submodule of V and therefore V = Rv. The map R + V, r H r v is a surjective R-homomorphism.
Ii) V is artinian and completely reducible. (iii) V is artinian and J ( V )= 0. Proof. 11. (ii)=+(iii). 11. Hence, by the definition of J ( V ) , J ( V ) = 0. 1, V is finitely cogenerated. ,V, = 0 for some maximal submodules V1,. . ,V, of V. 10(ii). 12. 2. Corollary. Let R be a ring. T h e n the following conditions are equivalent: Preliminaries 26 (i) RR is a finite direct s u m of irreducible modules. (ii) RR is completely reducible. (iii) R i s semisimple artinian. (iv) Every R-module i s completely reducible.
P ro o f. 15(iii). 7. 6. Group algebras Let R be a commutative ring and let G be a group, possibly infinite. Group algebras 45 The group algebra RG is the free R-module on the elements of G , with multiplication induced by that in G. More explicitly, RG consists of all formal linear combinations with finitely many zg # 0 subject to (i) (iv) c zg g , zg E R, g E G, C zg - g = C yg - g if and only if zg = yg for all g E G r(C zg - g ) = C(rzg)- g for all r E R It is easy to verify that these operations define RG as an associative R-algebra with 1 = 1 ~ -where 1 ~ 1~ and 1~ are identity elements of R and G, respectively.
Clifford Theory for Group Representations by Gregory Karpilovsky