View source for Brauer homomorphism

The Brauer homomorphism was first used by R. Brauer [[#References|[a1]]], [[#References|[a2]]], and developed further by H. Nagao [[#References|[a4]]] to establish a connection between the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b1108601.png" />-blocks of a [[Finite group|finite group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b1108602.png" /> and of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b1108603.png" />-local subgroups (cf. [[Block|Block]]). Subsequently, it has been used in a variety of other contexts within modular representation theory (cf. also [[Finite group, representation of a|Finite group, representation of a]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b1108604.png" /> be a [[Field|field]] of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b1108605.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b1108606.png" /> act by conjugation on the [[Group algebra|group algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b1108607.png" /> with fixed points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b1108608.png" />. The primitive idempotents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b1108609.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086010.png" /> correspond to the blocks <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086011.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086012.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086014.png" /> as a direct sum of indecomposable two-sided ideals.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086015.png" /> are subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086017.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086018.png" />-module, there is a relative trace mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086019.png" />, given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086021.png" /> runs through a set of coset representatives of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086022.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086023.png" />. One writes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086024.png" /> for the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086025.png" />. Applying this to the conjugation action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086026.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086027.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086028.png" /> is a defect group for the [[Block|block]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086029.png" /> (cf. also [[Defect group of a block|Defect group of a block]]) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086030.png" /> lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086031.png" /> but not in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086032.png" /> for any proper subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086033.png" />. The defect groups of a block form a single conjugacy class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086034.png" />-subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086035.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086036.png" /> is any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086037.png" />-subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086038.png" />, still acting on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086039.png" /> by conjugation, then there is a decomposition

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086040.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086041.png" />, and the sums of orbits of length greater than one are transfers from proper subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086042.png" />. The Brauer homomorphism

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is the projection onto the first factor in the above decomposition. Restricting to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086044.png" />, one obtains the homomorphism

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originally considered by Brauer.

Brauer's first main theorem states that the Brauer homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086046.png" /> establishes a one-to-one correspondence between the blocks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086047.png" /> with defect group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086048.png" /> and the blocks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086049.png" /> with defect group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086050.png" />. To make sense of this, first observe that every idempotent in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086051.png" /> lies in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086052.png" /> (because the other summand <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086053.png" /> is contained in the [[Jacobson radical|Jacobson radical]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086054.png" />). A block idempotent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086055.png" /> has defect group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086056.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086058.png" />. In the commutative diagram

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086059.png" /></td> </tr></table>

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086061.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086063.png" />-algebra. L. Puig [[#References|[a5]]], [[#References|[a6]]] has developed an extensive theory of  "interior G-algebras" , where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086064.png" /> 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 [[#References|[a7]]]. Further discussion of the Brauer homomorphism in this context can be found in [[#References|[a3]]] and [[#References|[a8]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Brauer,   "Zur Darstellungstheorie der Gruppen endlicher Ordnung"  ''Math. Z.'' , '''63'''  (1956)  pp. 406–441</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Brauer,   "Zur Darstellungstheorie der Gruppen endlicher Ordnung II"  ''Math. Z.'' , '''72'''  (1959)  pp. 25–46</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  B. Külshammer,   "Lectures on block theory" , ''London Math. Soc. Lecture Notes Ser.'' , '''161''' , Cambridge Univ. Press  (1991)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Nagao,   "A proof of Brauer's theorem on generalized decomposition numbers"  ''Nagoya Math. J.'' , '''22'''  (1963)  pp. 73–77</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  L. Puig,   "Pointed groups and construction of characters"  ''Math. Z.'' , '''170'''  (1981)  pp. 265–292</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  L. Puig,   "Pointed groups and construction of modules"  ''J. Algebra'' , '''116'''  (1988)  pp. 7–129</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  L. Puig,   "Nilpotent groups and their source algebras"  ''Invent. Math.'' , '''93'''  (1988)  pp. 77–116</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J. Thévenaz,   "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110860/b11086065.png" />-Algebras and modular representation theory" , ''Math. Monographs'' , Oxford Univ. Press  (1995)</TD></TR></table>