Brauer's main theorems are three theorems in
representation theory of finite groups linking the
blocks of a
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
(in characteristic ''p'') with those of its
''p''-local subgroups, that is to say, the
normalizers of its non-trivial ''p''-subgroups.
The second and third main theorems allow refinements of orthogonality relations for
ordinary characters which may be applied in finite
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
. 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'' either to ''0'' or to an idempotent element. In the latter case,
the idempotent 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.
Brauer's first main theorem
Brauer's first main theorem states that if
is a finite group and
is a
-subgroup of
, then there is a
bijection
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
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 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 character
Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field ''K'' of positive characteristic ''p'', necessarily a prime number. As well as h ...
s 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 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'').
References
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* gives a detailed proof of the Brauer's main theorems.
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Walter Feit
Walter Feit (October 26, 1930 – July 29, 2004) was an Austrian-born American mathematician who worked in finite group theory and representation theory. His contributions provided elementary infrastructure used in algebra, geometry, topology, n ...
, ''The representation theory of finite groups.'' North-Holland Mathematical Library, 25. North-Holland Publishing Co., Amsterdam-New York, 1982. xiv+502 pp. {{ISBN, 0-444-86155-6
Representation theory of finite groups
Theorems in representation theory