Modular representation theory: Difference between revisions

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{{Short description|Studies linear representations of finite groups over fields of positive characteristic}}

'''Modular representation theory''' is a branch of [[mathematics]], and thatis 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 groupsgroup]]s 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 (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==

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 ==

Finding a representation of the [[cyclic group]] of two elements over '''F'''₂ is equivalent to the problem of finding [[matrix (mathematics)|matrices]] whose square is the [[identity matrix]]. Over every field of characteristic other than 2, there is always a [[basis (linear algebra)|basis]] such that the matrix can be written as a [[diagonal matrix]] with only 1 or −1−1 occurring on the diagonal, such as

:<math>

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</math>

Over an algebraically closed field of positive characteristic, the representation theory of a finite cyclic group is fully explained by the theory of the [[Jordan normal form]]. Non-diagonal Jordan forms occur when the characteristic divides the order of the group.

the representation theory of a finite cyclic group is fully explained

by the theory of the [[Jordan normal form]]. Non-diagonal Jordan

forms occur when the characteristic divides the order of the group.

== Ring theory interpretation ==

Given a field ''K'' and a finite group ''G'', the [[group ring|group algebra]] ''K''[''G''] (which is the ''K''-[[vector space]] with ''K''-basis consisting of the elements of ''G'', endowed with algebra multiplication by extending the multiplication of ''G'' by linearity) is an [[Artinian ring]].

with ''K''-basis consisting of the elements of ''G'', endowed with

algebra multiplication by extending the multiplication

of ''G'' by linearity) is an [[Artinian ring]].

When the order of ''G'' is divisible by the characteristic of ''K'', the group algebra is not [[Semisimple algebraic group|semisimple]], hence has non-zero [[Jacobson radical]]. In that case, there are finite-dimensional modules for the group algebra that are not [[projective module]]s. By contrast, in the characteristic 0 case every [[irreducible representation]] is a [[direct summand]] of the [[regular representation]], hence is projective.

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characteristic ''p'' representation theory, ordinary character theory and structure of ''G'', especially as the latter relates to the embedding of, and relationships between, its ''p''-subgroups. Such results can be applied in [[group theory]] to problems not directly phrased in terms of representations.

Brauer introduced the notion now known as the '''Brauer character'''. When ''K'' is algebraically closed of positive characteristic ''p'', there is a bijection between roots of unity in ''K'' and complex roots of unity of order primecoprime to ''p''. Once a choice of such a bijection is fixed, the Brauer character of a representation assigns to each group element of order coprime to ''p'' the sum of complex roots of unity corresponding to the eigenvalues (including multiplicities) of that element in the given representation.

The Brauer character of a representation determines its composition

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==Reduction (mod ''p'')==

In the theory initially developed by Brauer, the link between ordinary representation theory and modular representation theory is best exemplified by considering the

[[group ring|group algebra]] of the group ''G'' over a complete [[discrete valuation ring]] ''R'' with [[residue field]] ''K'' of positive

valuation ring ''R'' with residue field ''K'' of positive

characteristic ''p'' and field of fractions ''F'' of characteristic

0, such as the [[p-adic number#p-adic integers|''p''-adic integers]]. The structure of ''R''[''G''] is closely related both to

the structure of the group algebra ''K''[''G''] and to the structure of the semisimple group algebra ''F''[''G''], and there is much interplay

between the module theory of the three algebras.

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

and, by a process often known informally as '''reduction (mod p)''',

arises by extension of scalars from an ''R''[''G'']-module.{{citation needed|date=August 2024}} In general, however, not all ''K''[''G'']-modules arise as reductions (mod ''p'') of

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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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 maximalvaluation orderring ''R'' of ''F''. The block corresponding to the primitive idempotent

''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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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, [[Everett C. Dade|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.