Brauer's three main theorems
Three results in the representation theory of finite groups
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Through Theorem, Characteristic (algebra) and Local property
The second and third main theorems allow refinements of orthogonality relations for ordinary characters which may be applied in finite group theory. These do not presently admit a proof purely in terms of ordinary characters. All three main theorems are stated in terms of the Brauer correspondence.
Brauer correspondence
There are many ways to extend the definition which follows, but this is close to the early treatments by Brauer. Let G be a finite group, p be a prime, F be a field of characteristic p. Let H be a subgroup of G which contains
for some p-subgroup Q of G, and is contained in the normalizer
- ,
where is the centralizer of Q in G.
The Brauer homomorphism (with respect to H) is a linear map from the center of the group algebra of G over F to the corresponding algebra for H. Specifically, it is the restriction to of the (linear) projection from to whose kernel is spanned by the elements of G outside . The image of this map is contained in , and it transpires that the map is also a ring homomorphism.
Since it is a ring homomorphism, for any block B of FG, the Brauer homomorphism sends the identity element of B to an idempotent element (possibly to 0). If the idempotent is non-zero, it may be decomposed as a sum of (mutually orthogonal) primitive idempotents of Z(FH). Each of these primitive idempotents is the multiplicative identity of some block of FH. The block b of FH is said to be a Brauer correspondent of B if its identity element occurs in this decomposition of the image of the identity of B under the Brauer homomorphism.
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Brauer's first main theorem
Brauer's first main theorem (Brauer 1944, 1956, 1970) states that if is a finite group and is a -subgroup of , then there is a bijection between the set of (characteristic p) blocks of with defect group and blocks of the normalizer with defect group D. This bijection arises because when , each block of G with defect group D has a unique Brauer correspondent block of H, which also has defect group D.
Brauer's second main theorem
Brauer's second main theorem (Brauer 1944, 1959) gives, for an element t whose order is a power of a prime p, a criterion for a (characteristic p) block of to correspond to a given block of , via generalized decomposition numbers. These are the coefficients which occur when the restrictions of ordinary characters of (from the given block) to elements of the form tu, where u ranges over elements of order prime to p in , are written as linear combinations of the irreducible Brauer characters of . The content of the theorem is that it is only necessary to use Brauer characters from blocks of which are Brauer correspondents of the chosen block of G.
Brauer's third main theorem
Brauer's third main theorem (Brauer 1964, theorem3) states that when Q is a p-subgroup of the finite group G, and H is a subgroup of G containing and contained in , then the principal block of H is the only Brauer correspondent of the principal block of G (where the blocks referred to are calculated in characteristic p).