On finite simple groups of essential dimension 3

0.07 MB·English

by  Arnaud Beauville

ON FINITE SIMPLE GROUPS OF ESSENTIAL DIMENSION 3
ARNAUDBEAUVILLE
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0 ABSTRACT. Weshowthattheonlyfinitesimplegroupsofessentialdimension3(over
2 C)areA6 andpossiblyPSL2(F11).Thisisaneasyconsequenceoftheclassification by

n Prokhorovofrationallyconnectedthreefoldswithanactionofasimplegroup.
a
J

8
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INTRODUCTION

]
G LetGbeafinitegroup,andX acomplexprojectivevarietywithafaithfulactionofG.
A WewillsaythatX isalinearizableifthereexistsacomplexrepresentationV ofGanda
rationaldominantG-equivariantmapV 99KX (suchamapiscalledacompressionofV).
.
h Theessential dimensioned(G)of G(overC)isthe minimaldimension of alllinearizable
t
a
G-varieties.Wehavetoreferto[BR]forthemotivationbehindthisdefinition;inavery
m
informalway,ed(G)istheminimumnumberofparametersneededtodefineallGalois
[
extensionsL/K withGaloisgroupGandK ⊃C.

2
Thegroupsofessentialdimension1arethecyclicgroupsandthediedralgroupD ,
n
v
2 nodd[BR].Thegroupsofessentialdimension2areclassifiedin[D2];thelistisalready
7 large,and such classification becomesprobablyintractable in higher dimension. How-
3 everthesimple(finite)groupsinthelistareonlyA andPSL (F ).Inthisnotewetryto
1 5 2 7
. goonestepfurther:
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0 Proposition. Thesimplegroupsofessentialdimension3areA andpossiblyPSL (F ).
1 6 2 11
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The result is an easy consequence of the remarkable paper of Prokhorov [P], who
:
v
classifiesallrationallyconnectedthreefoldsadmittingtheactionofasimplegroup.We
i
X canruleoutmostofthegroupsappearingin[P]thankstoasimplecriterion[RY]:ifaG-
r varietyXislinearizable,anyabeliansubgroupofGmustfixapointofX.Unfortunately
a
this criterion does not applyto PSL (F ), whose only abeliansubgroups are cyclic or
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isomorphicto(Z/2)2.
1. PROKHOROV’S LIST
Let G be a finite simple group with ed(G) = 3. By definition there exists a lineariz-
ableprojectiveG-threefoldX.ThisimpliesinparticularthatX isrationallyconnected.
Suchpairs(G,X)havebeenclassifiedin[P]:uptoconjugation, we havethe following
possibilities:
Date:January19,2011.
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2 ARNAUDBEAUVILLE
(1) G=SL (F )actingonaFanothreefoldX ⊂P8;
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(2) G=A ,A ,A ,PSL (F ),PSL (F ),orPSp (F ).
5 6 7 2 7 2 11 4 3
ThegroupsA ,A ,A haveessentialdimension2,3and4respectively,andPSL (F )
5 6 7 2 7
hasessentialdimension2[D1].WearenotabletosettlethecaseG=PSL (F )(see§3).
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AsforPSp (F ),wehave:
4 3
Proposition1. TheessentialdimensionofPSp(4,F )is4.
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Proof : The group Sp(4,F ) has a linear representation on the space W of functions on
3
F2, the Weil representation, for which we refer to [AR], Appendix I. This representation
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splitsasW = W+ ⊕W−,thespacesofevenandoddfunctions; wehavedimW+ = 5,
dimW− = 4. The central element (−I) of Sp(4,F ) acts on W by (−I)F (x) = F(−x),
3
henceitactstriviallyonW+,andas−IdonW−.Thuswegetafaithfulrepresentation
ofPSp(4,F )onW+,withacompressiontoP(W+)∼=P4,henceed(PSp(4,F ))≤4.
3 3
To prove that we have equality, we observe1 that PSp(4,F ) contains a subgroup
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isomorphic to (Z/2)4. One way to see this is to use the isomorphism PSp(4,F ) ∼=
3
SO+(5,F ): the group of diagonal matrices with entries ±1 and determinant 1 is con-
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tainedinSO+(5,F ),andisomorphicto(Z/2)4.By[BR]wehave
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ed(PSp(4,F ))≥ed((Z/2)4)=4.
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2. THE GROUPSL2(F8)
Itremainstoprovethatthepair(SL (F ),X)mentionedin(1)isnotlinearizable.To
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dothiswewillusethefollowingcriterion([RY],Appendix):
Lemma1. If(G,X)islinearizable,everyabeliansubgroupofGhasafixedpointinX.
Proposition2. TheessentialdimensionofSL (F )is≥4.
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ThegroupSL (F )hasarepresentationofdimension7,henceitsessentialdimension
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is≤6 –wedonotknowitsprecisevalue.
Proof : The group SL (F ) acts on a rational Fano threefold X ⊂ P8 in the following
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way[P].LetU beanirreducible9-dimensionalrepresentationofSL (F );thereexistsa
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non-degenerate invariant quadratic form q on U, unique up to a scalar. Then SL (F )
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acts on the orthogonal Grassmannian G (4,U) of 4-dimensional isotropic subspaces
iso
of U. This Grassmannian admits a O(q)-equivariant embedding into P15, given by the
half-spinor representation [M]. The threefold X is the intersection of G (4,U) with a
iso
subspaceP8 ⊂P15invariantunderSL (F ).
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I a
LetN ⊂ SL (F )bethesubgroupofmatrices ,a ∈ F .WewillshowthatN
2 8 8
0 I!
hasnofixedpointinG (4,U),andthereforeinX.
iso
1IamindebtedtoA.Duncanforthisobservation.
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Letχ bethecharacteroftherepresentationU.Wehaveχ (n) = 1forn ∈ N,n 6= 1
U U
(see for instance [C], 2.7). It follows that U restricted to N is the sum of the regular
representationandthetrivialone;inotherwords,asaN-modulewehave
U =C2⊕ C ,
1 λ
λX∈Nˆ
λ6=1
whereC istheone-dimensionalrepresentationassociatedtothecharacterλ.Thesub-
λ
spacesC andC mustbeorthogonalforα6=β;sinceqisnon-degenerate,itsrestriction
α β
toeachC (λ6=1)andtoC2mustbenon-degenerate.
λ 1
Now any vector subspace L ⊂ U fixed by N must be the sum of some of the C ,
λ
for λ 6= 1, and of some subspace of C2; this implies that L cannot be isotropic as soon
1
as dimL ≥ 2. Hence N has no fixed point on G (4,U), and X is not linearizable by
iso
Lemma1.
3. ABOUTPSL2(F11)
TheWeilrepresentationW−ofSL (F )factorsthroughPSL (F ),henceprovidesa
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5-dimensional representationof the latter group; thus its essential dimension is 3 or 4.
Accordingto[P]therearetworationallyconnectedthreefoldswithanactionofPSL (F ),
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theKleincubicXk ⊂ P4 givenby X2X = 0andaFanothreefoldXa ⊂ P9 of
i∈Z/5 i i+1
degree14,birationaltoXk.The groupPSL (F )hasorder660 = 22.3.5.11;itsabelian
P 2 11
subgroupsarecyclic,exceptthe2-Sylowsubgroupswhichareisomorphic to(Z/2)2. A
finiteorderautomorphismofarationallyconnectedvarietyhasalwaysafixedpoint(for
instance by the holomorphic Lefschetzformula); one checkseasily that a 2-Sylowsub-
group of PSL (F ) has a fixed point on both Xk and Xa. So lemma 1 does not apply,
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andanotherapproachisneeded.
REFERENCES
[AR] A.Adler,S.Ramanan:Moduliofabelianvarieties.Lect.NotesinMath.1644,Springer-Verlag,Berlin
(1996).
[BR] J.Buhler,Z.Reichstein:Ontheessentialdimensionofafinitegroup.CompositioMath.106(1997),no.2,
159–179.
[C] M.Collins:Representationsandcharactersoffinitegroups.CambridgeUniversityPress(1990).
[D1] A.Duncan:EssentialDimensionsofA7andS7.Math.Res.Lett.17(2010),no.2,263–266.
[D2] A.Duncan:FiniteGroupsofEssentialDimension2.PreprintarXiv:0912.1644,toappearinComment.
Math.Helv.
[M] S.Mukai:Curvesandsymmetricspaces,I.Am.J.Math.117(1995),no.6,1627–1644.
[P] Y.Prokhorov:SimplefinitesubgroupsoftheCremonagroupofrank3.PreprintarXiv:0908.0678.
[RY] Z. Reichstein, B. Youssin:EssentialdimensionsofalgebraicgroupsandaresolutiontheoremforG-
varieties.WithanappendixbyJ.Kolla´randE.Szabo´.Canad.J.Math.52(2000),no.5,1018–1056.
LABORATOIREJ.-A.DIEUDONNE´,UMR6621DUCNRS,UNIVERSITE´DENICE,PARCVALROSE,F-06108
NICECEDEX2,FRANCE
E-mailaddress:[email protected]