Modular representation theory

Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite groups over a field K of positive 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, combinatorics and number theory.The earliest work on representation theory over finite fields is by 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 invariants of some finite groups. The systematic study of modular representations, when the characteristic p divides the order of the group, was started by Brauer (1935) and was continued by him for the next few decades.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, there is always a basis such that the matrix can be written as a diagonal matrix with only 1 or −1 occurring on the diagonal, such asGiven a field K and a finite group G, the group algebra K (which is the K-vector spacewith K-basis consisting of the elements of G, endowed withalgebra multiplication by extending the multiplicationof G by linearity) is an Artinian ring.Modular representation theory was developed by Richard Brauer from about 1940 onwards to study in greater depth the relationships between thecharacteristic 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.In the theory initially developed by Brauer, the link between ordinary representation theory and modular representation theory is best exemplified by considering thegroup algebra of the group G over a complete discretevaluation ring R with residue field K of positivecharacteristic p and field of fractions F of characteristic0, such as the p-adic integers. The structure of R is closely related both tothe structure of the group algebra K and to the structure of the semisimple group algebra F, and there is much interplaybetween the module theory of the three algebras.In ordinary representation theory, the number of simple modules k(G) is equal to the number of conjugacy classes of G. 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 modular representation theory, while Maschke's theorem does not holdwhen 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. When the field F has characteristic 0, or characteristic coprime to the group order, there is still such a decomposition of the group algebra F as a sum of blocks (one for each isomorphism type of simple module), but the situation is relatively transparent when F is sufficiently large: each block is a full matrix algebra over F, the endomorphism ring of the vector space underlying the associated simple module.In ordinary representation theory, every indecomposable module is irreducible, and so every module is projective. However, the simple modules with 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 the endomorphism algebra of the underlying vector space, a full matrix algebra. In that case, the block is said to have 'defect 0'. Generally, the structure of projective modules is difficult to determine.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 thesecond Brauer character is that of the socle of a non-isomorphic projective indecomposable, and 1if the second Brauer character is that of its own socle. The multiplicity of an ordinary irreduciblecharacter in the character of the lift of a projective indecomposable is equal to the numberof occurrences of the Brauer character of the socle of the projective indecomposable when the restriction of the ordinary character to p-regular elements is expressed as a sum of irreducible Brauer characters.The composition factors of the projective indecomposable modules may be calculated as follows:Given the ordinary irreducible and irreducible Brauer characters of a particular finite group, the irreducible ordinary characters may be decomposed as non-negative integer combinations of the irreducible Brauer characters. 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 the transpose of D with D itselfresults in the Cartan matrix, usually denoted C; this is a symmetric matrix such that the entries in its j-th row are the multiplicities of the respective simple modules as compositionfactors of the j-th projective indecomposable module. The Cartanmatrix is non-singular; in fact, its determinant is a power of thecharacteristic of K.To each block B of the group algebra K, Brauer associated a certain p-subgroup, known as its defect group (where p is the characteristic of K). Formally, it is the largest p-subgroupD of G for which there is a Brauer correspondent of B for thesubgroup D C G ( D ) {displaystyle DC_{G}(D)}  , where C G ( D ) {displaystyle C_{G}(D)}   is the centralizer of D in G.