Difference between revisions of "Alperin conjecture"
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''Alperin weight conjecture'' | ''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 | + | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a1105301.png" /> is a prime number, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a1105303.png" />-local subgroup of a finite group is the normalizer (cf. also [[Normalizer of a subset|Normalizer of a subset]]) of a non-trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a1105304.png" />-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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a1105305.png" />-locally controlled. |
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a1105306.png" /> be an [[Algebraically closed field|algebraically closed field]] of prime characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a1105307.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a1105308.png" /> be a [[Finite group|finite group]]. Then the [[Group algebra|group algebra]] has a unique decomposition in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a1105309.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053010.png" />'s are mutually annihilating indecomposable two-sided ideals (cf. [[Ideal|Ideal]]). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053011.png" />'s are known as blocks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053012.png" /> (cf. also [[Block|Block]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053013.png" /> denote the number of isomorphism types of simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053014.png" />-modules, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053015.png" />. It is sometimes the case that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053016.png" /> is isomorphic to a full matrix algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053017.png" />. In that case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053018.png" /> is a block of defect zero of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053019.png" />. Such blocks have special significance; they are in bijection with the isomorphism types of projective simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053020.png" />-modules. Blocks other than blocks of defect zero are said to have positive defect. | Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a1105306.png" /> be an [[Algebraically closed field|algebraically closed field]] of prime characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a1105307.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a1105308.png" /> be a [[Finite group|finite group]]. Then the [[Group algebra|group algebra]] has a unique decomposition in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a1105309.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053010.png" />'s are mutually annihilating indecomposable two-sided ideals (cf. [[Ideal|Ideal]]). The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053011.png" />'s are known as blocks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053012.png" /> (cf. also [[Block|Block]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053013.png" /> denote the number of isomorphism types of simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053014.png" />-modules, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053015.png" />. It is sometimes the case that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053016.png" /> is isomorphic to a full matrix algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053017.png" />. In that case, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053018.png" /> is a block of defect zero of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053019.png" />. Such blocks have special significance; they are in bijection with the isomorphism types of projective simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053020.png" />-modules. Blocks other than blocks of defect zero are said to have positive defect. | ||
− | A weight of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053021.png" /> is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053023.png" /> is a (possibly trivial) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053024.png" />-subgroup (cf. also [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053025.png" />-group]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053027.png" /> is a projective simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053028.png" />-module. Weights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053030.png" /> are deemed to be equivalent if for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053031.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053033.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053034.png" />-modules. The | + | A weight of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053021.png" /> is a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053023.png" /> is a (possibly trivial) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053024.png" />-subgroup (cf. also [[P-group|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053025.png" />-group]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053027.png" /> is a projective simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053028.png" />-module. Weights <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053030.png" /> are deemed to be equivalent if for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053031.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053033.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053034.png" />-modules. The "non-blockwise" version of Alperin's weight conjecture simply asserts that the number of isomorphism types of simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053035.png" />-modules should equal the number of equivalence classes of weights of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053036.png" />. The number of equivalence classes of weights of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053037.png" /> is just the number of isomorphism types of projective simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053038.png" />-modules, and the number of equivalence classes of weights of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053039.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053040.png" /> is determined within <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053041.png" />-local subgroups. |
− | To describe the | + | 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053042.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053043.png" />-subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053044.png" />, the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053045.png" /> with kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053046.png" /> restricts to an algebra homomorphism from the fixed-point subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053047.png" /> (under conjugation by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053048.png" />) onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053049.png" />. Given a weight<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053050.png" />, one may view <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053051.png" /> as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053052.png" />-module. One assigns the weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053053.png" /> to the block <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053054.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053055.png" /> does not annihilate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053056.png" />. The "blockwise" version of Alperin's weight conjecture predicts that for each block <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053057.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053059.png" /> should equal the number of equivalence classes of weights assigned to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053060.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053061.png" /> is of defect <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053062.png" />, this is evidently true. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053063.png" /> is of positive defect, then the conjecture makes a prediction that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053064.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053065.png" />-locally determined in a precise manner, since no weights of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053066.png" /> are then assigned to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053067.png" />. |
Another interpretation of Alperin's weight conjecture has been given by R. Knörr and G.R. Robinson. Given a block <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053068.png" />, Brauer showed how to assign complex irreducible characters to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053069.png" />, and the number of such characters assigned to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053070.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053071.png" />, denoted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053072.png" />. | Another interpretation of Alperin's weight conjecture has been given by R. Knörr and G.R. Robinson. Given a block <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053068.png" />, Brauer showed how to assign complex irreducible characters to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053069.png" />, and the number of such characters assigned to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053070.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053071.png" />, denoted <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a110/a110530/a11053072.png" />. | ||
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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> |
Revision as of 17:31, 31 March 2012
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; 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). One of the central themes of Brauer's work was that many representation-theoretic invariants of finite groups should be "locally" determined. When is a prime number, a
-local subgroup of a finite group is the normalizer (cf. also Normalizer of a subset) of a non-trivial
-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
-locally controlled.
Let be an algebraically closed field of prime characteristic
, and let
be a finite group. Then the group algebra has a unique decomposition in the form
, where the
's are mutually annihilating indecomposable two-sided ideals (cf. Ideal). The
's are known as blocks of
(cf. also Block). Let
denote the number of isomorphism types of simple
-modules, and
. It is sometimes the case that
is isomorphic to a full matrix algebra over
. In that case,
is a block of defect zero of
. Such blocks have special significance; they are in bijection with the isomorphism types of projective simple
-modules. Blocks other than blocks of defect zero are said to have positive defect.
A weight of is a pair
, where
is a (possibly trivial)
-subgroup (cf. also
-group) of
and
is a projective simple
-module. Weights
and
are deemed to be equivalent if for some
one has
and
as
-modules. The "non-blockwise" version of Alperin's weight conjecture simply asserts that the number of isomorphism types of simple
-modules should equal the number of equivalence classes of weights of
. The number of equivalence classes of weights of the form
is just the number of isomorphism types of projective simple
-modules, and the number of equivalence classes of weights of the form
with
is determined within
-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. When is a
-subgroup of
, the projection
with kernel
restricts to an algebra homomorphism from the fixed-point subalgebra
(under conjugation by
) onto
. Given a weight
, one may view
as an
-module. One assigns the weight
to the block
if and only if
does not annihilate
. The "blockwise" version of Alperin's weight conjecture predicts that for each block
of
,
should equal the number of equivalence classes of weights assigned to
. If
is of defect
, this is evidently true. If
is of positive defect, then the conjecture makes a prediction that
is
-locally determined in a precise manner, since no weights of the form
are then assigned to
.
Another interpretation of Alperin's weight conjecture has been given by R. Knörr and G.R. Robinson. Given a block , Brauer showed how to assign complex irreducible characters to
, and the number of such characters assigned to
is
, denoted
.
Given a chain of strictly increasing
-subgroups of
, say
, one sets
, and
. Then one sets
, which is a sum of certain blocks of the group algebra
. Then the blockwise version of Alperin's weight conjecture is equivalent to the assertion that whenever
is a block of positive defect of
for some finite group
, then
![]() |
where denotes the collection of all such chains of
-subgroups of
(including the empty chain). Thus,
(the contribution to the alternating sum from the empty chain) is predicted to be
-locally controlled in a precise fashion.
J. Thévenaz has given a reformulation of Alperin's conjecture which is expressed in terms of equivariant -theory. E.C. Dade has continued the pattern of predictions of precise
-local control of representation-theoretic invariants by making a series of conjectures expressing the number of irreducible characters of defect
assigned to a block
of
as an alternating sum somewhat similar to that above. An irreducible character
is said to have defect
for the prime number
if
. 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 and offer the prospect of reducing these questions to questions about finite simple groups (cf. Simple finite group).
References
[a1] | J.L. Alperin, "Weights for finite groups" , Proc. Symp. Pure Math. , 47 , Amer. Math. Soc. (1987) pp. 369–379 MR0933373 Zbl 0657.20013 |
[a2] | E.C. Dade, "Counting characters in blocks I" Invent. Math. , 109 (1992) pp. 187–210 MR1168370 Zbl 0738.20011 |
[a3] | E.C. Dade, "Counting characters in blocks II" J. Reine Angew. Math. , 448 (1994) pp. 97–190 MR1266748 Zbl 0790.20020 |
[a4] | R. Knörr, G.R. Robinson, "Some remarks on a conjecture of Alperin" J. London Math. Soc. (2) , 39 (1989) pp. 48–60 MR0989918 Zbl 0672.20005 |
[a5] | J. Thévenaz, "Equivariant ![]() |
Alperin conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Alperin_conjecture&oldid=18681