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Tito Omburo (talk | contribs) →Blocks and the structure of the group algebra: valuation ring, not maximal order in the sense of the article (a more general sense). |
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{{Short description|
'''Modular representation theory''' is a branch of [[mathematics]], and is the part of [[representation theory]] that studies [[linear representation]]s of [[finite group]]s over a [[field (mathematics)|field]] ''K'' of positive [[characteristic (algebra)|characteristic]] ''p'', necessarily a [[prime number]]. As well as having applications to [[group theory]], modular representations arise naturally in other branches of mathematics, such as [[algebraic geometry]], [[coding theory]]{{Citation needed|reason=unveriviable and unsufficient citation about the source|date=May 2017}}, [[combinatorics]] and [[number theory]].
Within finite group theory, [[character theory|character-theoretic]] results proved by [[Richard Brauer]] using modular representation theory played an important role in early progress towards the [[classification of finite simple groups]], especially for [[simple group]]s whose characterization was not amenable to purely group-theoretic methods because their [[Sylow's theorems|Sylow 2-subgroup]]s were too small in an appropriate sense. Also, a general result on embedding of elements of [[order (group theory)|order]] 2 in finite groups called the [[Z* theorem]], proved by [[George Glauberman]] using the theory developed by Brauer, was particularly useful in the classification program.
If the characteristic ''p'' of ''K'' does not divide the [[order (group theory)|order]] |''G''|, then modular representations are completely reducible, as with ''ordinary'' (characteristic 0) representations, by virtue of [[Maschke's theorem]]. In the other case, when |''G''| ≡ 0 (mod ''p''), the process of averaging over the group needed to prove Maschke's theorem breaks down, and representations need not be completely reducible. Much of the discussion below implicitly assumes that the field ''K'' is sufficiently large (for example, ''K'' [[algebraically closed field|algebraically closed]] suffices), otherwise some statements need refinement.
==History==
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To obtain the blocks, the identity element of the group ''G'' is decomposed as a sum of primitive [[idempotent]]s
in ''Z''(''R''[G]), the [[center (ring theory)|center]] of the group algebra over the
''e'' is the two-sided ideal ''e'' ''R''[''G'']. For each indecomposable ''R''[''G'']-module, there is only one such primitive idempotent that does not annihilate it, and the module is said to belong to (or to be in) the corresponding block (in which case, all its [[composition factor]]s also belong to that block). In particular, each simple module belongs to a unique block. Each ordinary irreducible character may also be assigned to a unique block according to its decomposition as a sum of irreducible Brauer characters. The block containing the [[trivial representation|trivial module]] is known as the '''principal block'''.
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