Modular representation theory: Difference between revisions

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'''Modular representation theory''' is a branch of [[mathematics]], and that 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 groups whose characterization was not amenable to purely group-theoretic methods because their [[Sylow's theorems|Sylow 2-subgroupssubgroup]]s were too small in an appropriate sense. Also, a general result on embedding of elements of 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 |''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]] suffices), otherwise some statements need refinement.
 
==History==
 
The earliest work on representation theory over [[finite fieldsfield]]s is by {{harvtxt|Dickson|1902}} who showed that when ''p'' does not divide the order of the group, the representation theory is similar to that in characteristic 0. He also investigated [[modular invariant of a group|modular invariants]] of some finite groups. The systematic study of modular representations, when the characteristic ''p'' divides the order of the group, was started by {{harvtxt|Brauer|1935}} and was continued by him for the next few decades.
 
== Example ==
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Each ''R''[''G'']-module naturally gives rise to an ''F''[''G'']-module,
and, by a process often known informally as '''reduction (mod ''p'')''',
to a ''K''[''G'']-module. On the other hand, since ''R'' is a
[[principal ideal ___domain]], each finite-dimensional ''F''[''G'']-module
arises by extension of scalars from an ''R''[''G'']-module. In general,
however, not all ''K''[''G'']-modules arise as reductions (mod ''p'') of
''R''[''G'']-modules. Those that do are '''liftable'''.
 
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Each projective indecomposable module (and hence each projective module) in positive characteristic ''p'' may be lifted to a module in characteristic 0. Using the ring ''R'' as above, with residue field ''K'', the identity element of ''G'' may be decomposed as a sum of mutually orthogonal primitive [[idempotent]]s (not necessarily
central) of ''K''[''G'']. Each projective indecomposable ''K''[''G'']-module is isomorphic to ''e''.''K''[''G''] for a primitive idempotent ''e'' that occurs in this decomposition. The idempotent ''e'' lifts to a primitive idempotent, say ''E'', of ''R''[''G''], and the left module ''E''.''R''[''G''] has reduction (mod ''p'') isomorphic to ''e''.''K''[''G''].
 
==Some orthogonality relations for Brauer characters==
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Brauer's first main theorem states that the number of blocks of a finite group that have a given ''p''-subgroup as defect group is the same as the corresponding number for the normalizer in the group of that ''p''-subgroup.
 
The easiest block structure to analyse with non-trivial defect group is when the latter is cyclic. Then there are only finitely many isomorphism types of indecomposable modules in the block, and the structure of the block is by now well understood, by virtue of work of Brauer, [[E.C. Dade]], J.A. Green and [[John Griggs Thompson|J.G. Thompson]], among others. In all other cases, there are infinitely many isomorphism types of indecomposable modules in the block.
 
Blocks whose defect groups are not cyclic can be divided into two types: tame and wild. The tame blocks (which only occur for the prime 2) have as a defect group a [[dihedral group]], [[semidihedral group]] or (generalized) [[quaternion group]], and their structure has been broadly determined in a series of papers by [[Karin Erdmann]]. The indecomposable modules in wild blocks are extremely difficult to classify, even in principle.