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''Alperin weight conjecture''

Modular representation theory of finite groups is the study of representations of groups over fields of finite characteristic (cf. also [[Finite group, representation of a|Finite group, representation of a]]; [[Finite group|Finite group]]; [[Field|Field]]). This theory was first developed by R. Brauer, who was motivated largely by a wish to obtain information about complex characters of finite groups (cf. [[Character of a group|Character of a group]]). One of the central themes of Brauer's work was that many representation-theoretic invariants of finite groups should be "locally" determined. When  $  p $
is a prime number, a  $  p $-
local subgroup of a finite group is the normalizer (cf. also [[Normalizer of a subset|Normalizer of a subset]]) of a non-trivial  $  p $-
subgroup of that group. An important feature of Alperin's weight conjecture is that it makes a precise prediction as to how a fundamental representation-theoretic invariant should be  $  p $-
locally controlled.

Let  $  F $
be an [[Algebraically closed field|algebraically closed field]] of prime characteristic  $  p $, 
and let  $  G $
be a [[Finite group|finite group]]. Then the [[Group algebra|group algebra]] has a unique decomposition in the form  $  FG = B _ {1} \oplus \dots \oplus B _ {m} $, 
where the  $  B _ {i} $'
s are mutually annihilating indecomposable two-sided ideals (cf. [[Ideal|Ideal]]). The  $  B _ {i} $'
s are known as blocks of  $  FG $(
cf. also [[Block|Block]]). Let  $  l ( B _ {i} ) $
denote the number of isomorphism types of simple  $  B _ {i} $-
modules, and  $  l ( FG ) = \sum _ {i = 1 }   ^ {m} l ( B _ {i} ) $. 
It is sometimes the case that  $  B _ {i} $
is isomorphic to a full matrix algebra over  $  F $. 
In that case,  $  B _ {i} $
is a block of defect zero of  $  FG $. 
Such blocks have special significance; they are in bijection with the isomorphism types of projective simple  $  FG $-
modules. Blocks other than blocks of defect zero are said to have positive defect.

A weight of  $  FG $
is a pair  $  ( Q,S ) $, 
where  $  Q $
is a (possibly trivial)  $  p $-
subgroup (cf. also [[P-group| $  p $-
group]]) of  $  G $
and  $  S $
is a projective simple  $  FN _ {G} ( Q ) /Q $-
module. Weights  $  ( Q,S ) $
and  $  ( R,T ) $
are deemed to be equivalent if for some  $  x \in G $
one has  $  Q  ^ {x} = R $
and  $  S  ^ {x} \cong T $
as  $  FN _ {G} ( R ) /R $-
modules. The "non-blockwise" version of Alperin's weight conjecture simply asserts that the number of isomorphism types of simple  $  FG $-
modules should equal the number of equivalence classes of weights of  $  FG $. 
The number of equivalence classes of weights of the form  $  ( \{ 1 _ {G} \} ,S ) $
is just the number of isomorphism types of projective simple  $  FG $-
modules, and the number of equivalence classes of weights of the form  $  ( Q,S ) $
with  $  Q \neq \{ 1 _ {G} \} $
is determined within  $  p $-
local subgroups.

To describe the "blockwise" version of the conjecture one has to assign weights to blocks. This is achieved by means of the [[Brauer homomorphism|Brauer homomorphism]]. When  $  Q $
is a  $  p $-
subgroup of  $  G $, 
the projection  $  { { \mathop{\rm br} } _ {Q} } : {FG } \rightarrow {FC _ {G} ( Q ) } $
with kernel  $  F [ G \backslash C _ {G} ( Q ) ] $
restricts to an algebra homomorphism from the fixed-point subalgebra  $  FG  ^ {Q} $(
under conjugation by  $  Q $) 
onto  $  FC _ {G} ( Q ) $. 
Given a weight $  ( Q,S ) $, 
one may view  $  S $
as an  $  FN _ {G} ( Q ) $-
module. One assigns the weight  $  ( Q,S ) $
to the block  $  B $
if and only if  $  1 _ {B} { \mathop{\rm br} } _ {Q} $
does not annihilate  $  S $. 
The "blockwise" version of Alperin's weight conjecture predicts that for each block  $  B $
of  $  FG $, 
$  l ( B ) $
should equal the number of equivalence classes of weights assigned to  $  B $. 
If  $  B $
is of defect  $  0 $, 
this is evidently true. If  $  B $
is of positive defect, then the conjecture makes a prediction that  $  l ( B ) $
is  $  p $-
locally determined in a precise manner, since no weights of the form  $  ( \{ 1 _ {G} \} , S ) $
are then assigned to  $  B $.

Another interpretation of Alperin's weight conjecture has been given by R. Knörr and G.R. Robinson. Given a block  $  B $, 
Brauer showed how to assign complex irreducible characters to  $  B $, 
and the number of such characters assigned to  $  B $
is  $  { \mathop{\rm dim} } _ {F} ( Z ( B ) ) $, 
denoted  $  k ( B ) $.

Given a chain  $  \sigma $
of strictly increasing  $  p $-
subgroups of  $  G $, 
say  $  \sigma = Q _ {1} < Q _ {2} < \dots < Q _ {n} $, 
one sets  $  G _  \sigma  = \cap _ {i = 1 }   ^ {n} N _ {G} ( Q _ {i} ) $, 
and  $  | \sigma | = n $. 
Then one sets  $  B _  \sigma  = ( 1 _ {B} { \mathop{\rm br} } _ {Q _ {n}  } ) \cdot FG _  \sigma  $, 
which is a sum of certain blocks of the group algebra  $  FG _  \sigma  $. 
Then the blockwise version of Alperin's weight conjecture is equivalent to the assertion that whenever  $  B $
is a block of positive defect of  $  FG $
for some finite group  $  G $, 
then

$$ 
\sum _ {\sigma \in \Delta _ {p} /G } ( -1 ) ^ {\left | \sigma \right | } k ( B _  \sigma  ) = 0,
$$

where  $  \Delta _ {p} $
denotes the collection of all such chains of  $  p $-
subgroups of  $  G $(
including the empty chain). Thus,  $  k ( B ) $ 
(the contribution to the alternating sum from the empty chain) is predicted to be  $  p $-
locally controlled in a precise fashion.

J. Thévenaz has given a reformulation of Alperin's conjecture which is expressed in terms of equivariant [[K-theory| $  K $-
theory]]. E.C. Dade has continued the pattern of predictions of precise  $  p $-
local control of representation-theoretic invariants by making a series of conjectures expressing the number of irreducible characters of defect  $  d $
assigned to a block  $  B $
of  $  FG $
as an alternating sum somewhat similar to that above. An irreducible character  $  \chi $
is said to have defect  $  d $
for the prime number  $  p $
if  $  p  ^ {d} \| { {| G | } / {\chi ( 1 ) } } $. 
Dade's conjectures may be viewed as unifying and extending Alperin's weight conjecture and the Alperin–McKay conjecture. Furthermore, they are compatible with techniques of [[Clifford theory|Clifford theory]] and offer the prospect of reducing these questions to questions about finite simple groups (cf. [[Simple finite group|Simple finite group]]).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.L. Alperin, "Weights for finite groups" , ''Proc. Symp. Pure Math.'' , '''47''' , Amer. Math. Soc. (1987) pp. 369–379 {{MR|0933373}} {{ZBL|0657.20013}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> E.C. Dade, "Counting characters in blocks I" ''Invent. Math.'' , '''109''' (1992) pp. 187–210 {{MR|1168370}} {{ZBL|0738.20011}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> E.C. Dade, "Counting characters in blocks II" ''J. Reine Angew. Math.'' , '''448''' (1994) pp. 97–190 {{MR|1266748}} {{ZBL|0790.20020}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> R. Knörr, G.R. Robinson, "Some remarks on a conjecture of Alperin" ''J. London Math. Soc. (2)'' , '''39''' (1989) pp. 48–60 {{MR|0989918}} {{ZBL|0672.20005}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> J. Thévenaz, "Equivariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053099.png" />-theory and Alperin's conjecture" ''J. Pure Appl. Algebra'' , '''85''' (1993) pp. 185–202 {{MR|1207508}} {{ZBL|}} </TD></TR></table>