Brauer–Nesbitt theorem

In mathematics, the Brauer–Nesbitt theorem can refer to several different theorems proved by Richard Brauer and Cecil J. Nesbitt in the representation theory of finite groups.

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. Moreover it belongs to a block of defect zero. A block of defect zero contains only one ordinary character and only one modular character.

Another version states that if k is a field of characteristic zero, A is a k-algebra, V, W are semisimple A-modules which are finite dimensional over k, and Tr_V = Tr_W as elements of Hom_k(A,k), then V and W are isomorphic as A-modules.

References

• Curtis, Reiner, Representation theory of finite groups and associative algebras, Wiley 1962.
• Brauer, R.; Nesbitt, C. On the modular characters of groups. Ann. of Math. (2) 42, (1941). 556-590.