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- 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 TrV = TrW as elements of Homk(A,k), then V and W are isomorphic as A-modules. Let be a group and be some field. If are two finite-dimensional semisimple representations such that the characteristic polynomials of and coincide for all , then and are isomorphic representations. If or , then the condition on the characteristic polynomials can be changed to the condition that the traces of and coincide for all . As a consequence, let be a semisimple (continuous) -adic representations of the absolute Galois group of some field , unramified outside some finite set of primes . Then the representation is uniquely determined by the values of the traces of for (also using the Chebotarev density theorem). (en)
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- 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. (en)
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- Brauer–Nesbitt theorem (en)
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