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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.
Richard Dagobert Brauer (February 10, 1901 – April 17, 1977) was a German and American mathematician. He worked mainly in abstract algebra , but made important contributions to number theory . He was the founder of modular representation theory .
The situation with representations over a field of positive characteristic, so-called "modular representations", is more delicate, but Richard Brauer developed a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters of modular representations.
Brauer laid out a list of problems that help to define representation theory.Number 23 is the Height Zero Conjecture, in which Brauer claims that abelian groups must have a particular quality ...
The Brauer Height Zero Conjecture is a conjecture in modular representation theory of finite groups relating the degrees of the complex irreducible characters in a Brauer block and the structure of its defect groups. It was formulated by Richard Brauer in 1955.
Cecil James Nesbitt, Ph.D., F.S.A., M.A.A.A. (1912–2001) was a mathematician who was a Ph.D. student of Richard Brauer and wrote many influential papers in the early history of modular representation theory. Nesbitt taught actuarial mathematics at the University of Michigan from 1938 to 1980. Nesbitt was born in Ontario, Canada.
It follows, using some results of Richard Brauer from modular representation theory, that the prime divisors of the orders of the elements of each conjugacy class of a finite group can be deduced from its character table (an observation of Graham Higman).
In modular representation theory, the Brauer–Nesbitt theorem on blocks of defect zero states that a character whose order is divisible by the highest power of a prime p dividing the order of a finite group remains irreducible when reduced mod p and vanishes on all elements whose order is divisible by p.