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 that part of [[representation theory]] that studies [[linear representation]]s of [[finite group]] ''G'' over a [[field (mathematics)|field]] ''K'' of positive [[characteristic (algebra)|characteristic]]. As well as having applications to group theory, modular representations arise

'''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 groupsgroup]]s whose characterization was not amenable to purely group-theoretic methods because their [[Sylow's 2theorems|Sylow subgroups2-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.

Within finite [[group 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 2 subgroups 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 of(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.

(characteristic 0) representations, by virtue of [[Maschke's theorem]]. The proof of Maschke's theorem relies on being able to divide by the group order, which is not meaningful when the order of ''G'' is divisible by the characteristic of ''K''. In that case, representations need not be

completely reducible, unlike the ordinary (and the coprime characteristic) case. 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 then, 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.

The systematic study of modular representations, when the characteristic divides the order of the group, was started by {{harvtxt|Brauer|1935}} and 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

factors but not, in general, its equivalence type. The irreducible

Brauer characters are those afforded by the simple modules.

These are integral ( though not necessarily non-negative) combinations

of the restrictions to elements of order coprime to ''p'' of the ordinary irreducible

characters. Conversely, the restriction to the elements of order primecoprime to ''p'' of

each ordinary irreducible character is uniquely expressible as a non-negative

integer combination of irreducible Brauer characters.

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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|''0p''-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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In modular representation theory, while Maschke's theorem does not hold

when the characteristic divides the group order, the group algebra may be decomposed as the direct sum of a maximal collection of two-sided ideals known as '''blocks'''. (whenWhen the field ''KF'' has characteristic ''0'', or characteristic coprime to the group order, there is alsostill such a decomposition of the group algebra ''KF''[''G''] as a sum of blocks (one for each isomorphism type of simple module), but the situation is relatively transparent (at least when ''KF'' is sufficiently large): each block is a full matrix algebra over ''KF'', the endomorphism ring of the vector space underlying the associated simple module).

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'''.

== Projective modules ==

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non-isomorphic socles. The multiplicity of a projective indecomposable module as a summand of the group algebra (viewed as the regular module) is the dimension of its socle (for large enough fields of characteristic zero, this recovers the fact that each simple module occurs with multiplicity equal to its dimension as a direct summand of the regular module).

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==

When a projective module is lifted, the associated character vanishes on all elements of order divisible by ''p'', and (with consistent choice of roots of unity), agrees with the Brauer character of the original characteristic ''p'' module on ''p''-regular elements. The (usual character-ring) inner product of the Brauer character of a projective indecomposable with any other Brauer character can thus be defined: this is ''0'' if the

second Brauer character is that of the socle of a non-isomorphic projective indecomposable, and ''1''

if the second Brauer character is that of its own socle. The multiplicity of an ordinary irreducible

character in the character of the lift of a projective indecomposable is equal to the number

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To each block ''B'' of the group algebra ''K''[''G''], Brauer associated a certain ''p''-subgroup, known as its '''defect group''' (where ''p'' is the characteristic of ''K''). Formally, it is the largest ''p''-subgroup

''D'' of ''G'' for which there is a [[Brauer's three main theorems|Brauer correspondent]] of ''B'' for the

subgroup <math>DC_G(D)</math>, where <math>C_G(D)</math> is the [[centralizer]] of ''D'' in ''G''.

The defect group of a block is unique up to conjugacy and has a strong influence on the structure of the block. For example, if the defect group is trivial, then the block contains just one simple module, just one ordinary character, the ordinary and Brauer irreducible characters agree on elements of order prime to the relevant characteristic ''p'', and the simple module is projective. At the other extreme, when ''K'' has characteristic ''p'', the [[Sylow]] ''p''-subgroup of the finite group ''G'' is a defect group for the principal block of ''K''[''G''].

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multiplicity one. Also, the power of ''p'' dividing the index of the defect group of a block is the [[greatest common divisor]] of the powers of ''p'' dividing the dimensions of the simple modules in that block, and this coincides with the greatest common divisor of the powers of ''p'' dividing the degrees of the ordinary irreducible characters in that block.

Other relationships between the defect group of a block and character theory include Brauer's result that if no conjugate of the ''p''-part of a group element ''g'' is in the defect group of a given block, then each irreducible character in that block vanishes at ''g''. This is a one of many consequences of Brauer's second main theorem.

The defect group of a block also has several characterizations in the more module-theoretic approach to block theory, building on the work of [[Sandy Green (mathematician)|J. A. Green]], which associates a ''p''-subgroup

known as the '''vertex''' to an indecomposable module, defined in terms of '''relative projectivity''' of the module. For example, the vertex of each indecomposable module in a block is contained (up to conjugacy)

in the defect group of the block, and no proper subgroup of the defect group has that property.

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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.

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== References ==

*{{Citation | last1=Brauer | first1=R. | author1-link=Richard Brauer | title=Über die Darstellung von Gruppen in Galoisschen Feldern | url=httphttps://books.google.com/books?id=hkexAAAAIAAJ | publisher=Hermann et cie, |___location=Paris . | series=Actualités Scientifiques et Industrielles, | id=[http://projecteuclid.org/euclid.bams/1183499883 review] | year=1935 | volume=195 | pages=1–15}}

*{{Citation | last1=Dickson | first1=Leonard Eugene | author1-link=Leonard Eugene Dickson | title=On the Group Defined for any Given Field by the Multiplication Table of Any Given Finite Group | jstor=1986379 | publisher=[[American Mathematical Society]] | ___location=Providence, R.I. | year=1902 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=3 | issue=3 | pages=285–301 | doi=10.2307/1986379| doi-access=free }}

* {{cite book | author=[[Jean-Pierre Serre]] | title=Linear Representations of Finite Groups | url=https://archive.org/details/linearrepresenta1977serr | url-access=registration | publisher=[[Springer-Verlag]] | year=1977 | isbn=0-387-90190-6| author-link=Jean-Pierre Serre }}

* {{cite book | author=Walter Feit | authorlink=Walter Feit | title=The representation theory of finite groups | series=North-Holland Mathematical Library | volume=25 | publisher=North-Holland Publishing | ___location=Amsterdam-New York | year=1982 | isbn=0-444-86155-6 }}