In mathematics, modular representation theory is the branch of representation theory that studies linear representations of finite group G over a field K such that the characteristic of K is non-zero. An example of modular representation theory would be the study of representations of the cyclic group of two elements over F₂, the field with two elements. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, a finite group is a group which has finitely many elements. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na...

If the characteristic of K does not divide the order of G then modular representations are similar to characteristic zero representations. In these cases, Maschke's theorem yields that every representation is a direct sum of irreducible representations. The key step in the proof of Maschke's theorem is to average over the elements of the group, which fails when the order of G is divisible by the characteristic of K. In this case, the modular case, the representation theory is quite different from the characteristic 0 case, called the ordinary representation case. In particular, representations need not be direct sums of irreducible representations, which is always true in the ordinary case. In mathematics, in particular group representation theory, Maschkes theorem is the basic result proving that linear representations of a finite group over the complex numbers break up into irreducible pieces. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, the term irreducible is used in several ways. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, the term irreducible is used in several ways. ...

Example

Finding a representation of the cyclic group of two elements over F₂ is equivalent to the problem of finding matrices whose square is the identity matrix. Over every field of characteristic other than 2, we can always find a basis such that the matrix can be written as a diagonal matrix with only 1 or −1 occurring on the diagonal, such as In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na...

Over F₂, we can find many other possible matrices, such as

Ring theory interpretation

In terms of ring theory, the group algebra In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In the theory of group representations, the group algebra is any of various constructions to assign to a group (either a locally compact topological group, or a group without a topology, i. ...

K[G]

is not a semisimple ring in the case when the order of G is divisible by the characteristic of K, thus it will have a Jacobson radical that is non-zero. This also implies that there will exist finite-dimensional modules for the group algebra which are not projective modules. By contrast, in the characteristic 0 case every irreducible representation is a direct summand in the regular representation, implying that it is projective. In mathematics, the term semisimple is used in a number of related ways, within different subjects. ... In ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal of R which contains those elements of R which in a sense are close to zero. // The Jacobson radical is denoted by J(R) and can be defined in the following equivalent... In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module (that is, a module with basis vectors). ... In mathematics, the term irreducible is used in several ways. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, and in particular the theory of group representations, the regular representation of a group G is the linear representation afforded by the group action of G on itself. ...

The group algebra is an Artinian ring. Modular representation theory was developed by Richard Brauer from about 1940 onwards to provide more detailed information linked to the structure of G. Such results are applied in group theory to problems not directly phrased in terms of representations. In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. ... Richard Dagobert Brauer (February 10, 1901 - April 17, 1977) was a leading German and American mathematician. ... Group theory is that branch of mathematics concerned with the study of groups. ...

Number of simple modules

In ordinary representation theory, the number of simple modules k(G) is equal to the number of conjugacy classes. In the modular case, the number l(G) of simple modules is equal to the number of conjugacy classes whose elements have order coprime to the relevant prime p, the so-called p-regular classes. In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure. ...

Structure of the group algebra

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. Each indecomposable module then belongs to a unique block, and all its composition factors belong to that block. In particular, each simple module lies in 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 module is known as the principal block. In mathematics, a composition series of a group G is a normal series such that each Hi is a maximal proper normal subgroup of Hi+1. ... In mathematics, in particular group representation theory, a group representation of the group G is called a trivial representation if It is defined on a one-dimensional vector space V over a field K. All elements g of G act on V as the identity mapping. ...

Projective modules

In ordinary representation theory, every indecomposable module is irreducible, and so every module is projective. However, the simple modules in characteristic dividing the group order are rarely projective. Indeed, if a simple module is projective, then it is the only simple module in its block, which is then isomorphic to its endomorphism ring. In that case, the block is said to have 'defect 0'. Generally, the structure of projective modules is difficult to determine.

For a finite group, the projective indecomposable modules are in a one-to-one correspondence with the simple modules: the socle of each projective indecomposable is simple (and isomorphic to the top), and this affords the bijection, as non-isomorphic projective indecomosables have non-isomorphic socles. The multiplicity of a projective indecomposable module as a summand of the the group algebra is the dimension of its socle ( for large enough fields of characteristic zero, this recovers the fact that each irreducible representation occurs with multiplicity equal to its dimension as a direct summand of the regular module). Look up Socle in Wiktionary, the free dictionary. ...

The composition factors of the projective indecomposable modules may be calculated as follows: In mathematics, a composition series of a group G is a normal series such that each Hi is a maximal proper normal subgroup of Hi+1. ...

Given the ordinary irreducible and irreducible Brauer characters of a particular finite group, one can decompose the irreducible ordinary characters as linear combinations of the irreducible Brauer characters. The coefficients which occur are always non-negative integers. The integers involved can be placed in a matrix, with the ordinary irreducible characters assigned rows and the irreducible Brauer characters assigned columns. This is referred to as the decomposition matrix, and is frequently labelled D. It is customary to place the trivial ordinary and Brauer characters in the first row and column respectively. The product of D with its transpose results in the Cartan matrix, usually denoted C; this is a symmetric matrix whose entries are the multiplicities of the respective simple modules as composition factors of the projective indecomposable modules. In mathematics, more specifically in group representation, the character of a group representation is a function which associates to each element of the group an element of the field of the representation space. ... In mathematics, and in particular modular representation theory, a decomposition matrix is a matrix that results from writing the irreducible ordinary characters in terms of the irreducible modular characters, where the entries of the two sets of characters are taken to be over all conjugacy classes of elements of order... In mathematics, the term Cartan matrix has two meanings. ...

Defect groups

To each block of a finite group, a certain p-subgroup, the defect group, is attached. This defect group 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 Braur irreducible characters agree on elements of order prime to p, and the simple module is projective.

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 also coincides with the greatest common divisor of the powers of p dividing the degrees of the ordinary irreducible characters in that block. By a theorem of R.Brauer, the number of blocks of a group which 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. In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf), of two non-zero integers, is the largest positive integer that divides both numbers without remainder. ...

The easiest case to analyse when the defect group is non-trivial is when it is a cyclic group. Then there are only finitely many indecomposable modules lying in the block, and the structure of the block is relatively easy to understand. In all other cases, there are infinitely many indecomposable modules lying in a block. In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na...

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 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. This article may be confusing for some readers, and should be edited to enhance clarity. ... In mathematics, the quasidihedral groups (also known as semidihedral groups) are groups with similar properties to the dihedral groups. ... Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). ...