Brauer second main theorem
For notation and definitions, see Brauer first main theorem.
Let 
be an element of 
whose order is a power of 
. The 
-section of 
associated to 
is the set of all elements of 
whose 
-part is conjugate to 
. Brauer second main theorem relates the values of irreducible characters of 
on the 
-section associated to 
to values of characters in certain blocks of 
.
Suppose that 
is an irreducible character of 
(cf. also Character of a group), afforded by the 
-free right 
-module 
, and belonging to the block 
(cf. also Defect group of a block). Let 
be a 
-element of 
, and let 
. For all 
-subgroups 
of 
, 
; hence 
is defined for all blocks 
of 
. One can organize the block decomposition of 
as 
. Let 
be the projection of 
on 
, and let 
be the projection of 
on 
. The restriction 
of 
to 
can be decomposed as 
. If 
is the character of 
and 
is the character of 
, then of course 
for all 
. Brauer's second main theorem states that for all elements 
of order prime to 
, 
. Thus, the values of 
on the 
-section associated to 
are determined in the blocks of 
sent to 
by the Brauer correspondence (cf. also Brauer first main theorem).
This theorem was first proved in [a1]. See also [a2], [a3], and [a4].
References
| [a1] | R. Brauer, "Zur Darstellungstheorie der Gruppen endlicher Ordnung II" Math. Z. , 72 (1959) pp. 22–46 |
| [a2] | C. Curtis, I. Reiner, "Methods of representation theory" , II , Wiley (1987) |
| [a3] | W. Feit, "The representation theory of finite groups" , North-Holland (1982) |
| [a4] | H. Nagao, Y. Tsushima, "Representation of finite groups" , Acad. Press (1987) |
Brauer second main theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brauer_second_main_theorem&oldid=18936