By Peter Schneider (auth.)
Representation idea stories maps from teams into the final linear crew of a finite-dimensional vector house. For finite teams the idea is available in certain flavours. within the 'semisimple case' (for instance over the sphere of advanced numbers) it is easy to use personality conception to totally comprehend the representations. This by way of a long way isn't really adequate while the attribute of the sector divides the order of the group.
Modular illustration idea of finite Groups includes this moment scenario. Many extra instruments are wanted for this situation. to say a few, there's the systematic use of Grothendieck teams resulting in the Cartan matrix and the decomposition matrix of the crowd in addition to Green's direct research of indecomposable representations. there's additionally the tactic of writing the class of all representations because the direct fabricated from convinced subcategories, the so-called 'blocks' of the gang. Brauer's paintings then establishes correspondences among the blocks of the unique crew and blocks of yes subgroups the philosophy being that one is thereby diminished to a less complicated state of affairs. specifically, possible degree how nonsemisimple a class a block is by way of the dimensions and constitution of its so-called 'defect group'. a lot of these techniques are made particular for the instance of the specific linear team of two-by-two matrices over a finite major field.
Although the presentation is strongly biased in the direction of the module theoretic standpoint an test is made to strike a undeniable stability through additionally displaying the reader the gang theoretic technique. particularly, relating to illness teams an in depth facts of the equivalence of the 2 techniques is given.
This e-book goals to familiarize scholars on the masters point with the elemental effects, instruments, and methods of an attractive and critical algebraic concept. a few uncomplicated algebra including the semisimple case are assumed to be identified, even though all proof for use are restated (without proofs) within the textual content. in a different way the publication is completely self-contained.
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Additional info for Modular Representation Theory of Finite Groups
1 The Setting 45 homomorphism cG : K0 k[G] −→ Rk (G) [P ] −→ [P ]. Hence, so far, there is the diagram of homomorphisms RK (G) Rk (G) cG κ ρ K0 (R[G]) K0 (k[G]). Clearly, R[G] is an R-algebra which is finitely generated as an R-module. Let us collect some of what we know in this situation. 6) R[G] is left and right noetherian, and any finitely generated R[G]-module is complete as well as R[G]πR -adically complete. 7) The Krull–Remak–Schmidt theorem holds for any finitely generated R[G]-module. 5) 1 ∈ R[G] can be written as a sum of pairwise orthogonal primitive idempotents; the set of all central idempotents in R[G] is finite; 1 is equal to the sum of all primitive idempotents in Z(R[G]); any R[G]-module has a block decomposition.
4 there are the corresponding homomorphisms between Grothendieck groups K0 (K[G]) [P ]−→[K⊗R P ] κ ρ K0 (R[G]) [P ]−→[P /πR P ] K0 (k[G]). For the vertical arrow observe that, quite generally for any R[G]-module M, we have K[G] ⊗R[G] M = K ⊗R R[G] ⊗R[G] M = K ⊗R M. We put RK (G) := R K[G] and Rk (G) := R k[G] . iii. On the other hand, as a finite-dimensional k-vector space the group ring k[G] of course is left and right artinian. 1 The Setting 45 homomorphism cG : K0 k[G] −→ Rk (G) [P ] −→ [P ].
2 The Triangle 47 Definition Let V be a finite-dimensional K-vector space; a lattice L in V is an R-submodule L ⊆ V for which there exists a K-basis e1 , . . , ed of V such that L = Re1 + · · · + Red . Obviously, any lattice is free as an R-module. Furthermore, with L also aL, for any a ∈ K × , is a lattice in V . 1 i. Let L be an R-submodule of a K-vector space V ; if L is finitely generated then L is free. ii. Let L ⊆ V be an R-submodule of a finite-dimensional K-vector space V ; if L is finitely generated as an R-module and L generates V as a K-vector space then L is a lattice in V .