Brauer homomorphism
The Brauer homomorphism was first used by R. Brauer [a1], [a2], and developed further by H. Nagao [a4] to establish a connection between the -blocks of a finite group
and of the
-local subgroups (cf. Block). Subsequently, it has been used in a variety of other contexts within modular representation theory (cf. also Finite group, representation of a).
Let be a field of characteristic
, and let
act by conjugation on the group algebra
with fixed points
. The primitive idempotents
in
correspond to the blocks
of
, so that
and
as a direct sum of indecomposable two-sided ideals.
If are subgroups of
and
is a
-module, there is a relative trace mapping
, given by
, where
runs through a set of coset representatives of
in
. One writes
for the image of
. Applying this to the conjugation action of
on
, one says that
is a defect group for the block
(cf. also Defect group of a block) if
lies in
but not in
for any proper subgroup
. The defect groups of a block form a single conjugacy class of
-subgroups of
.
If is any
-subgroup of
, still acting on
by conjugation, then there is a decomposition
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as a direct sum of a subring and a two-sided ideal. This is because the left-hand side has as a basis the orbit sums. The orbits of length one span , and the sums of orbits of length greater than one are transfers from proper subgroups of
. The Brauer homomorphism
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is the projection onto the first factor in the above decomposition. Restricting to , one obtains the homomorphism
![]() |
originally considered by Brauer.
Brauer's first main theorem states that the Brauer homomorphism establishes a one-to-one correspondence between the blocks of
with defect group
and the blocks of
with defect group
. To make sense of this, first observe that every idempotent in
lies in
(because the other summand
is contained in the Jacobson radical
). A block idempotent
has defect group
if and only if
and
. In the commutative diagram
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the top- and right-hand mappings are surjective, and hence so is the bottom mapping. It therefore induces the bijection of block idempotents described in the first main theorem.
In subsequent work, the Brauer homomorphism has been generalized to arbitrary modules:
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The concept is especially useful if is a
-algebra. L. Puig [a5], [a6] has developed an extensive theory of "interior G-algebras" , where
is acting via inner automorphisms, and in which the Brauer homomorphism plays a central role. His work has had extensive impact both on module theory and on block theory, especially his classification of nilpotent blocks [a7]. Further discussion of the Brauer homomorphism in this context can be found in [a3] and [a8].
References
[a1] | R. Brauer, "Zur Darstellungstheorie der Gruppen endlicher Ordnung" Math. Z. , 63 (1956) pp. 406–441 |
[a2] | R. Brauer, "Zur Darstellungstheorie der Gruppen endlicher Ordnung II" Math. Z. , 72 (1959) pp. 25–46 |
[a3] | B. Külshammer, "Lectures on block theory" , London Math. Soc. Lecture Notes Ser. , 161 , Cambridge Univ. Press (1991) |
[a4] | H. Nagao, "A proof of Brauer's theorem on generalized decomposition numbers" Nagoya Math. J. , 22 (1963) pp. 73–77 |
[a5] | L. Puig, "Pointed groups and construction of characters" Math. Z. , 170 (1981) pp. 265–292 |
[a6] | L. Puig, "Pointed groups and construction of modules" J. Algebra , 116 (1988) pp. 7–129 |
[a7] | L. Puig, "Nilpotent groups and their source algebras" Invent. Math. , 93 (1988) pp. 77–116 |
[a8] | J. Thévenaz, "![]() |
Brauer homomorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brauer_homomorphism&oldid=16983