Editing Modular representation theory (section)

==Brauer characters==
Modular representation theory was developed by [[Richard Brauer]] from about 1940 onwards to study in greater depth the relationships between the
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 coprime 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 coprime to ''p'' of
each ordinary irreducible character is uniquely expressible as a non-negative
integer combination of irreducible Brauer characters.