New PDF release: Modular Representation Theory of Finite Groups

By Peter Schneider (auth.)

ISBN-10: 1447148312

ISBN-13: 9781447148319

ISBN-10: 1447148320

ISBN-13: 9781447148326

Representation concept reports maps from teams into the final linear workforce of a finite-dimensional vector house. For finite teams the idea is available in precise flavours. within the 'semisimple case' (for instance over the sphere of complicated numbers) you'll use personality idea to totally comprehend the representations. This by means of a long way isn't really adequate while the attribute of the sphere divides the order of the group.

Modular illustration thought of finite Groups contains this moment scenario. Many extra instruments are wanted for this example. to say a few, there's the systematic use of Grothendieck teams resulting in the Cartan matrix and the decomposition matrix of the gang in addition to Green's direct research of indecomposable representations. there's additionally the method of writing the class of all representations because the direct fabricated from definite subcategories, the so-called 'blocks' of the crowd. Brauer's paintings then establishes correspondences among the blocks of the unique staff and blocks of convinced subgroups the philosophy being that one is thereby diminished to an easier state of affairs. particularly, you can still degree how nonsemisimple a class a block is by means of the scale and constitution of its so-called 'defect group'. these kind of techniques are made specific for the instance of the particular linear staff of two-by-two matrices over a finite top field.

Although the presentation is strongly biased in the direction of the module theoretic standpoint an try is made to strike a undeniable stability via additionally displaying the reader the crowd theoretic process. particularly, when it comes to disorder teams a close evidence of the equivalence of the 2 methods is given.

This ebook goals to familiarize scholars on the masters point with the elemental effects, instruments, and methods of an attractive and significant algebraic concept. a few uncomplicated algebra including the semisimple case are assumed to be recognized, even supposing all proof for use are restated (without proofs) within the textual content. another way the publication is fullyyt self-contained.

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10 we find a projective cover P M −→ M of M as an A-module such that ∼ = P M / Jac(A)P M −→ M. 10 shows that P M in fact is a finitely generated A-module. 8 the isomorphism class {P M } only depends on the isomorphism class {M}. We conclude that Z[MA ] −→ Z[MA ] {M} −→ {P M } is a well-defined homomorphism. If N is a second A-module of finite length with g projective cover P N − → N as above then f ⊕g P M ⊕ P N −−−→ M ⊕ N is surjective with (P M ⊕ P N )/ Jac(A)(P M ⊕ P N ) = P M / Jac(A)P M ⊕ P N / Jac(A)P N ∼ = M ⊕ N.

Before we can establish the finer properties of the Cartan–Brauer triangle we need to develop the theory of induction. 3 The Ring Structure of RF (G), and Induction In this section we let F be an arbitrary field, and we consider the group ring F [G] and its Grothendieck group RF (G) := R(F [G]). 3 The Ring Structure of RF (G), and Induction 55 Let V and W be two (finitely generated) F [G]-modules. The group G acts on the tensor product V ⊗F W by g(v ⊗ w) := gv ⊗ gw for v ∈ V and w ∈ W. In this way V ⊗F W becomes a (finitely generated) F [G]-module, and we obtain the multiplication map Z[MF [G] ] × Z[MF [G] ] −→ Z[MF [G] ] {V }, {W } −→ {V ⊗F W }.

Let e ∈ R be any idempotent. i we must have e ∈ R × . Multiplying the identity e2 = e by e−1 gives e = 1. Now let us assume, vice versa, that 1 is the only idempotent in R. 7, the factor ring R := R/ Jac(R) also has no other idempotent than 1. 1, the R-module L := R is indecomposable. 2, is of finite length. 4 implies that EndR (L) is a local ring. ). We obtain that R and R are local rings. Since Jac(R) = Jac(R/ Jac(R)) = {0} the ring R in fact is a skew field. 1 implies that R is local. 12 Suppose that R is an R0 -algebra, which is finitely generated as an R0 -module, over a noetherian complete commutative ring R0 such that R0 / Jac(R0 ) is artinian; then the map set of all central idempotents in R e −→ −→ set of all central idempotents in R/ Jac(R0 )R e := e + Jac(R0 )R is bijective; moreover, this bijection satisfies: – e, f are orthogonal if and only if e, f are orthogonal; – e is primitive in Z(R) if and only if e is primitive in Z(R/ Jac(R0 )R).

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Modular Representation Theory of Finite Groups by Peter Schneider (auth.)

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