Stably Cayley semisimple groups

0.3 MB·

by  Mikhail Borovoi

1
4
Stably Cayley semisimple groups
1
0
2
Mikhail Borovoi and Boris Kunyavski˘ı
p
e
S

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1

]
G
Abstract. A linear algebraic group G over a field k is called a
A
CayleygroupifitadmitsaCayleymap,i.e.,aG-equivariantbirational
. isomorphism over k between the group variety G and its Lie algebra
h
t Lie(G). A prototypical example is the classical “Cayley transform”
a
for the special orthogonal group SO defined by Arthur Cayley in
m n
1846. A linear algebraic group G is called stably Cayley if G×S is
[
Cayleyforsomesplitk-torusS. WeclassifystablyCayleysemisimple

4 groups over an arbitrary field k of characteristic 0.
v
4
2010 Mathematics Subject Classification: 20G15, 20C10.
7
Keywords and Phrases: Linear algebraic group, stably Cayley group,
7
5 quasi-permutation lattice.
.
1
0
4 To Alexander Merkurjev on the occasion of his 60th birthday
1
:
v 0 Introduction
i
X
Letkbeafieldofcharacteristic0andk¯afixedalgebraicclosureofk. LetGbea
ar connectedlinearalgebraick-group. Abirationalisomorphismφ: G9≃9KLie(G)
iscalledaCayley mapifitisequivariantwithrespecttotheconjugationaction
ofGonitselfandtheadjointactionofGonitsLiealgebraLie(G),respectively.
A linear algebraic k-group G is called Cayley if it admits a Cayley map, and
stably Cayley if G× (G )r is Cayley for some r ≥ 0. Here G denotes
k m,k m,k
the multiplicative group over k. These notions were introduced by Lemire,
PopovandReichstein[LPR];foramoredetaileddiscussionandnumerousclas-
sical examples we refer the reader to [LPR, Introduction]. The main results
of[LPR]arethe classificationsofCayleyandstablyCayleysimple groupsover
an algebraically closed field k of characteristic 0. Over an arbitrary field k of
characteristic0stablyCayleysimplek-groups,stablyCayleysimplyconnected
semisimple k-groups and stably Cayley adjoint semisimple k-groups were clas-
sified in the paper [BKLR] of Borovoi, Kunyavski˘ı, Lemire and Reichstein. In
2 M. Borovoi and B. Kunyavski˘ı
thepresentpaper,buildingonresultsof[LPR]and[BKLR],weclassifyallsta-
bly Cayley semisimple k-groups (not necessarily simple, or simply connected,
or adjoint) over an arbitrary field k of characteristic 0.
Byasemisimple(orreductive)k-groupwealwaysmeanaconnectedsemisimple
(orreductive)k-group. Weshallneedthefollowingresultof[BKLR]extending
[LPR, Theorem 1.28].
Theorem 0.1 ([BKLR, Theorem1.4]). Let k be a field of characteristic 0 and
G an absolutely simple k-group. Then the following conditions are equivalent:
(a) G is stably Cayley over k;
(b) G is an arbitrary k-form of one of the following groups:
SL , PGL , PGL (n≥1), SO (n≥5), Sp (n≥1), G ,
3 2 2n+1 n 2n 2
or an inner k-form of PGL (n≥2).
2n
In this paper we classify stably Cayley semisimple groups overan algebraically
closed field k of characteristic 0 (Theorem 0.2) and, more generally, over an
arbitrary field k of characteristic0 (Theorem 0.3). Note that Theorem 0.2 was
conjectured in [BKLR, Remark 9.3].
Theorem 0.2. Let k be an algebraically closed field of characteristic 0 and G
a semisimple k-group. Then G is stably Cayley if and only if G decomposes
into a direct product G × ···× G of its normal subgroups, where each G
1 k k s i
(i = 1,...,s) either is a stably Cayley simple k-group (i.e., isomorphic to
one of the groups listed in Theorem 0.1) or is isomorphic to the stably Cayley
semisimple k-group SO .
4
Theorem 0.3. Let G be a semisimple k-group over a field k of characteristic
0 (not necessarily algebraically closed). Then G is stably Cayley over k if and
only if G decomposes into a direct product G × ···× G of its normal k-
1 k k s
subgroups, where each G (i = 1,...,s) is isomorphic to the Weil restriction
i
R G for some finite field extension l /k, and each G is either a stably
li/k i,li i i,li
Cayley absolutely simple groupover l (i.e., oneof thegroups listed in Theorem
i
0.1) or an l -form of the semisimple group SO (which is always stably Cayley,
i 4
but is not absolutely simple and can be not l -simple).
i
Note that the “if” assertionsin Theorems0.2 and0.3 follow immediately from
the definitions.
The rest of the paper is structured as follows. In Section 1 we recall the
definition of a quasi-permutation lattice and state some known results, in par-
ticular,anassertionfrom[LPR,Theorem1.27]thatreducesTheorem0.2toan
assertion on lattices. In Sections 2 and 3 we construct certain families of non-
quasi-permutationlattices. In particular,we correctan inaccuracyin [BKLR];
see Remark 2.5. In Section 4 we prove (in the language of lattices) Theorem
Stably Cayley semisimple groups 3
0.2inthespecialcasewhenGisisogenoustoadirectproductofsimplegroups
of type A with n ≥ 3. In Section 5 we prove (again in the language of
n−1
lattices) Theorem 0.2 in the general case. In Section 6 we deduce Theorem
0.3 from Theorem 0.2. In Appendix A we prove in terms of lattices only, that
certain quasi-permutation lattices are indeed quasi-permutation.
1 Preliminaries on quasi-permutation groups and on character
lattices
In this section we gather definitions and known results concerning quasi-
permutation lattices, quasi-invertible lattices and character lattices that we
need for the proofs of Theorems 0.2 and 0.3. For details see [BKLR, Sections
2 and 10] and [LPR, Introduction].
1.1. By a lattice we mean a pair (Γ,L) where Γ is a finite group acting on a
finitely generated free abelian group L. We say also that L is a Γ-lattice. A
Γ-lattice L is called a permutation lattice if it has a Z-basis permuted by Γ.
Following Colliot-Th´el`eneand Sansuc [CTS], we say that two Γ-lattices L and
L′ are equivalent, and write L∼L′, if there exist short exact sequences
0→L→E →P →0 and 0→L′ →E →P′ →0
with the same Γ-lattice E, where P and P′ are permutation Γ-lattices. For a
proofthat this isindeed anequivalence relationsee [CTS, Lemma8,p. 182]or
[Sw, Section 8]. Note that if there exists a short exact sequence of Γ-lattices
0→L→L′ →Q→0
where Q is a permutation Γ-lattice, then, taking in account the trivial short
exact sequence
0→L′ →L′ →0→0,
we obtain that L∼L′. If Γ-lattices L,L′,M,M′ satisfy L∼L′ and M ∼M′,
then clearly L⊕M ∼L′⊕M′.
Definition1.2. AΓ-latticeLis calledaquasi-permutation latticeifthereexists
a short exact sequence
0→L→P →P′ →0, (1.1)
where both P and P′ are permutation Γ-lattices.
Lemma 1.3 (well-known). A Γ-lattice L is quasi-permutation if and only if
L∼0.
Proof. If L is quasi-permutation, then sequence (1.1) together with the trivial
short exact sequence
0→0→P →P →0
4 M. Borovoi and B. Kunyavski˘ı
shows that L∼0. Conversely, if L∼0, then there are short exact sequences
0→L→E →P →0 and 0→0→E →P′ →0,
where P and P′ are permutation lattices. From the second exact sequence
we have E ∼= P′, hence E is a permutation lattice, and then the first exact
sequence shows that L is a quasi-permutation lattice.
Definition 1.4. A Γ-lattice L is calledquasi-invertible if it is a direct summand
of a quasi-permutation Γ-lattice.
Note that if a Γ-lattice L is not quasi-invertible, then it is not quasi-
permutation.
Lemma 1.5 (well-known). If a Γ-lattice L is quasi-permutation (resp., quasi-
invertible) and L′ ∼ L, then L′ is quasi-permutation (resp., quasi-invertible)
as well.
Proof. IfLisquasi-permutation,thenusingLemma1.3weseethatL′ ∼L∼0,
hence L′ is quasi-permutation. If L is quasi-invertible, then L⊕M is quasi-
permutationforsomeΓ-latticeM,andby Lemma1.3wehaveL⊕M ∼0. We
see that L′⊕M ∼ L⊕M ∼ 0, and by Lemma 1.3 we obtain that L′⊕M is
quasi-permutation, hence L′ is quasi-invertible.
Let Z[Γ] denote the group ring of a finite group Γ. We define the Γ-lattice J
Γ
by the exact sequence
N
0→Z−−−→Z[Γ]→J →0,
Γ
whereN isthenormmap,see[BKLR,beforeLemma10.4]. Wereferto[BKLR,
Proposition10.6]for a proofof the following result, due to Voskresenski˘ı[Vo1,
Corollary of Theorem 7]:
Proposition 1.6. Let Γ = Z/pZ×Z/pZ, where p is a prime. Then the Γ-
lattice J is not quasi-invertible.
Γ
Note that if Γ=Z/2Z×Z/2Z, then rankJ =3.
Γ
We shall use the following lemma from [BKLR]:
Lemma 1.7 ([BKLR, Lemma 2.8]). Let W ,...,W be finite groups. For each
1 m
i = 1,...,m, let V be a finite-dimensional Q-representation of W . Set V :=
i i
V ⊕ ···⊕ V . Suppose L ⊂ V is a free abelian subgroup, invariant under
1 m
W := W ×···×W . If L is a quasi-permutation W-lattice, then for each
1 m
i=1,...,m the intersection L :=L∩V is a quasi-permutation W -lattice.
i i i
Weshallneedthenotion,dueto[LPR]and[BKLR],ofthecharacterlatticeofa
reductivek-groupGoverafieldk. Letk¯beaseparableclosureofk. LetT ⊂G
be a maximal torus (defined over k). Set T = T × k¯, G = G× k¯. Let X(T)
k k
Stably Cayley semisimple groups 5
denote the character group of T := T × k¯. Let W = W(G,T) := N (T)/T
k G
denote the Weyl group, it acts on X(T). Consider the canonical Galois action
onX(T), itdefines a homomorphismGal(k¯/k)→AutX(T). The imageimρ⊂
AutX(T) normalizes W, hence imρ ·W is a subgroup of AutX(T). By the
character lattice of G we mean the pair X(G) := (imρ·W, X(T)) (up to an
isomorphism it does not depend on the choice of T). In particular, if k is
algebraically closed, then X(G)=(W,X(T)).
We shall reduce Theorem 0.2 to an assertion about quasi-permutation lattices
using the following result due to [LPR]:
Proposition 1.8 ([LPR, Theorem 1.27], see also [BKLR, Theorem 1.3]). A
reductivegroupGoveranalgebraically closedfieldk ofcharacteristic 0isstably
Cayley if and only if its character lattice X(G) is quasi-permutation, i.e., X(T)
is a quasi-permutation W(G,T)-lattice.
We shall use the following result due to Cortella and Kunyavski˘ı [CK] and to
Lemire, Popov and Reichstein [LPR].
Proposition 1.9 ([CK], [LPR]). Let D be a connected Dynkin diagram. Let
R=R(D) denote the corresponding root system, W =W(D) denote the Weyl
group, Q = Q(D) denote the root lattice, and P = P(D) denote the weight
lattice. WesaythatLisanintermediatelatticebetweenQandP ifQ⊂L⊂P
(note that L = Q and L = P are possible). Then the following list gives (up
to an isomorphism) all the pairs (D,L) such that L is a quasi-permutation
intermediate W(D)-lattice between Q(D) and P(D):
Q(A ), Q(B ), P(C ), X(SO ) (then D =D ),
n n n 2n n
or D is any connected Dynkin diagram of rank 1 or 2 (i.e. A , A , B , or
1 2 2
G ) and L is any lattice between Q(D) and P(D), (i.e., either L = P(D) or
2
L=Q(D)).
Proof. The positive result (the assertionthat the lattices in the list are indeed
quasi-permutation)followsfromtheassertionthatthecorrespondinggroupsare
stably Cayley (or that their generic toriare stably rational),see the references
in [CK], Section 3. See Appendix A below for a proof of this positive result in
termsoflatticesonly. ThedifficultpartofProposition1.9isthenegativeresult
(the assertion that all the other lattices are not quasi-permutation). This was
provedin [CK, Theorem0.1]in the cases when L=Q or L=P, and in [LPR,
Propositions 5.1 and 5.2] in the cases when Q(L(P (this can happen only
when D =A or D =D ).
n n
Remark 1.10. It follows from Proposition 1.9 that, in particular, the follow-
ing lattices are quasi-permutation: Q(A ), P(A ), P(A ), P(B ), Q(C ),
1 1 2 2 2
Q(G )=P(G ), Q(D )=Q(A ), X(SL /µ )=X(SO ).
2 2 3 3 4 2 6
6 M. Borovoi and B. Kunyavski˘ı
2 A family of non-quasi-permutation lattices
Inthissectionweconstructafamilyofnon-quasi-permutation(evennon-quasi-
invertible) lattices.
2.1. We consider a Dynkin diagram D⊔∆ (disjoint union). We assume that
D = D (a finite disjoint union), where each D is of type B (l ≥1) or
i∈I i i li i
D (l ≥ 2) (and where B = A , B = C , D = A ⊔A , and D = A
li Fi 1 1 2 2 2 1 1 3 3
are permitted). We denote by m the cardinality of the finite index set I. We
assume that ∆= µ ∆ (disjoint union), where ∆ is of type A , n ≥2
ι=1 ι ι 2nι−1 ι
(A =D is permitted). We assume that m≥1 and µ≥0 (in the case µ=0
3 3
F
the diagram ∆ is empty).
For each i ∈ I we realize the root system R(D ) of type B or D in the
i li li
standard way in the space Vi := Rli with basis (es)s∈Si where Si is an index
set consisting of l elements; cf. [Bou, Planche II] for B (l ≥ 2) (the relevant
i l
formulas for B are similar) and [Bou, Planche IV] for D (l ≥ 3) (again, the
1 l
relevantformulasforD aresimilar). LetM ⊂V denotethelatticegenerated
2 i i
by the basis vectors (e ) . Let P ⊃ M denote the weight lattice of the
s s∈Si i i
root system D . Set S = S (disjoint union). Consider the vector space
i i i
V = V with basis (e ) . Let M ⊂ V denote the lattice generated by
i i s sF∈S D
the basis vectors (e ) , then M = M . Set P = P .
L s s∈S D i i D i i
For each ι = 1,...,µ we realize the roLot system R(∆ι) oLf type A2nι−1 in the
standard way in the subspace V of vectors with zero sum of the coordinates
ι
in the space R2nι with basis ει,1,...,ει,2nι; cf. [Bou, Planche I]. Let Qι be the
rootlatticeofR(∆ )withbasisε −ε , ε −ε , ..., ε −ε ,and
ι ι,1 ι,2 ι,2 ι,3 ι,2nι−1 ι,2nι
let P ⊃Q be the weight lattice of R(∆ ). Set Q = Q , P = P .
ι ι ι ∆ ι ι ∆ ι ι
Set L L
µ µ
W := W(D )× W(∆ ), L′ =M ⊕Q = M ⊕ Q ,
i ι D ∆ i ι
i∈I ι=1 i∈I ι=1
Y Y M M
then W acts on L′ and on L′⊗ZR. For each i consider the vector
x = e ∈M ,
i s i
sX∈Si
then 1x ∈P . For each ι consider the vector
2 i i
ξ =ε −ε +ε −ε +···+ε −ε ∈Q ,
ι ι,1 ι,2 ι,3 ι,4 ι,2nι−1 ι,2nι ι
then 1ξ ∈P ; see [Bou, Planche I]. Write
2 ι ι
ξ′ =ε −ε , ξ′′ =ε −ε +···+ε −ε ,
ι ι,1 ι,2 ι ι,3 ι,4 ι,2nι−1 ι,2nι
then ξ =ξ′+ξ′′. Consider the vector
ι ι ι
µ µ
1 1 1 1
v = x + ξ = e + ξ ∈P ⊕P .
i ι s ι D ∆
2 2 2 2
i∈I ι=1 s∈S ι=1
X X X X
Stably Cayley semisimple groups 7
Set
L=hL′,vi, (2.1)
then [L:L′]=2 because v ∈ 1L′rL′. Note that the sublattice L⊂P ⊕P
2 D ∆
is W-invariant. Indeed, the groupW acts on (P ⊕P )/(M ⊕Q ) trivially.
D ∆ D ∆
Proposition 2.2. We assume that m ≥ 1, m+µ ≥ 2. If µ = 0, we assume
that not all of D are of types B or D . Then the W-lattice L as in (2.1) is
i 1 2
not quasi-invertible, hence not quasi-permutation.
Proof. We consider a group Γ = {e,γ ,γ ,γ } of order 4, where γ ,γ ,γ are
1 2 3 1 2 3
of order 2. The idea of our proof is to construct an embedding
j: Γ→W
in such a way that L, viewed as a Γ-lattice, is equivalent to its Γ-sublattice
L , and L is isomorphic to a direct sum of a Γ-sublattice L ≃ J of rank 3
1 1 0 Γ
and a number of Γ-lattices of rank 1. Since by Proposition1.6 J is not quasi-
Γ
invertible,thiswillimplythatL andLarenotquasi-invertibleΓ-lattices,and
1
hence L is not quasi-invertible as a W-lattice. We shall now fill in the details
of this argument in four steps.
Step1. WebeginbypartitioningeachS fori∈I intothree(non-overlapping)
i
subsets S , S and S , subject to the requirement that
i,1 i,2 i,3
|S |≡|S |≡|S |≡l (mod 2) if D is of type D . (2.2)
i,1 i,2 i,3 i i li
We then set U to be the union of the S , U to be the union of the S , and
1 i,1 2 i,2
U to be the union of the S , as i runs over I.
3 i,3
Lemma 2.3. (i) If µ≥1, the subsets S , S and S of S can be chosen,
i,1 i,2 i,3 i
subject to (2.2), so that U 6=∅.
1
(ii) If µ=0 (and m≥2), the subsets S , S and S of S can be chosen,
i,1 i,2 i,3 i
subject to (2.2), so that U ,U ,U 6=∅.
1 2 3
Toprovethelemma,firstconsidercase(i). ForallisuchthatD isoftypeD
i li
with odd l , we partition S into three non-empty subsets of odd cardinalities.
i i
For all the other i we take S = S , S =S = ∅. Then U 6= ∅ (note that
i,1 i i,2 i,3 1
m≥1) and (2.2) is satisfied.
Incase(ii),ifoneoftheD isoftypeD wherel ≥3isodd, thenwepartition
i li i
S for each such D into three non-empty subsets of odd cardinalities. We
i i
partition all the other S as follows:
i
S =S =∅ and S =S . (2.3)
i,1 i,2 i,3 i
Clearly U ,U ,U 6=∅ and (2.2) is satisfied.
1 2 3
IfthereisnoD oftypeD withoddl ≥3,butoneoftheD ,sayfori=i ,is
i li i i 0
D withevenl ≥4,thenwepartitionS intotwonon-emptysubsetsS and
l i0 i0,1
8 M. Borovoi and B. Kunyavski˘ı
S ofevencardinalities,andset S =∅. We partitionthe sets S for i6=i
i0,2 i0,3 i 0
as in (2.3) (note that by our assumption m ≥ 2). Once again, U ,U ,U 6= ∅
1 2 3
and (2.2) is satisfied.
If there is no D of type D with l ≥3 (odd or even), but one of the D , say
i li i i
fori=i ,isoftypeB withl ≥2,wepartitionS intotwonon-emptysubsets
0 l i0
S andS , andsetS =∅. We partitionthe sets S for i6=i asin (2.3)
i0,1 i0,2 i0,3 i 0
(again, note that m≥2). Once again, U ,U ,U 6=∅ and (2.2) is satisfied.
1 2 3
SincebyourassumptionnotallofD areoftypeB orD ,wehaveexhausted
i 1 2
all the cases. This completes the proof of Lemma 2.3.
Step 2. We continue proving Proposition 2.2. We construct an embedding
Γ֒→W.
For s ∈ S we denote by c the automorphism of L taking the basis vector
s
e to −e and fixing all the other basis vectors. For ι = 1,...,µ we define
s s
τ(12) = Transp((ι,1),(ι,2)) ∈ W (the transposition of the basis vectors ε
ι ι ι,1
and ε ). Set
ι,2
τ>2 =Transp((ι,3),(ι,4))· ··· ·Transp((ι,2n −1),(ι,2n ))∈W .
ι ι ι ι
Write Γ={e,γ ,γ ,γ } and define an embedding j: Γ֒→W as follows:
1 2 3
µ
j(γ )= c · τ(12)τ>2;
1 s ι ι
s∈YSrU1 ιY=1
µ
j(γ )= c · τ(12);
2 s ι
s∈YSrU2 ιY=1
µ
j(γ )= c · τ>2.
3 s ι
s∈YSrU3 ιY=1
Note that if D is of type D , then by (2.2) for κ = 1,2,3 the cardinality
i li
#(Si rSi,κ) is even, hence the product of cs over s ∈ Si rSi,κ is contained
in W(Di) for all such i, and therefore, j(γκ) ∈ W. Since j(γ1), j(γ2) and
j(γ ) commute, are of order 2, and j(γ )j(γ ) = j(γ ), we see that j is a
3 1 2 3
homomorphism. Ifµ≥1,then,since2n1 ≥4,clearlyj(γκ)6=1forκ =1,2,3,
hence j is an embedding. If µ = 0, then the sets S rU , S rU and S rU
1 2 3
are nonempty, and again j(γκ)6=1 for κ =1,2,3, hence j is an embedding.
Step 3. We construct a Γ-sublattice L of rank 3. Write a vector x∈L as
0
µ 2nι
x= b e + β ε ,
s s ι,ν ι,ν
s∈S ι=1ν=1
X XX
where b ,β ∈ 1Z. Set n′ = µ (n −1). Define a Γ-equivarianthomomor-
s ι,ν 2 ι=1 ι
phism
φ: L→Zn′, x7→P(β +β )
ι,2λ−1 ι,2λ ι=1,...,µ, λ=2,...,nι
Stably Cayley semisimple groups 9
(we skip λ=1). We obtain a short exact sequence of Γ-lattices
0→L →L−−φ−→Zn′ →0,
1
where L := kerφ. Since Γ acts trivially on Zn′, we have L ∼ L. Therefore,
1 1
it suffices to show that L is not quasi-invertible.
1
Recall that
µ
1 1
v = e + ξ .
s ι
2 2
s∈S ι=1
X X
Set v =γ ·v, v =γ ·v, v =γ ·v. Set
1 1 2 2 3 3
L =hv,v ,v ,v i.
0 1 2 3
We have
µ
1 1 1
v = e − e − ξ ,
1 s s ι
2 2 2
sX∈U1 s∈UX2∪U3 Xι=1
whence
v+v = e . (2.4)
1 s
sX∈U1
We have
µ
1 1 1
v = e − e + (−ξ′+ξ′′),
2 2 s 2 s 2 ι ι
sX∈U2 s∈UX1∪U3 Xι=1
whence
µ
v+v = e + ξ′′. (2.5)
2 s ι
sX∈U2 Xι=1
We have
µ
1 1 1
v = e − e + (ξ′−ξ′′),
3 2 s 2 s 2 ι ι
sX∈U3 s∈UX1∪U2 Xι=1
whence
µ
v+v = e + ξ′. (2.6)
3 s ι
sX∈U3 Xι=1
Clearly, we have
v+v +v +v =0.
1 2 3
Since the set {v,v ,v ,v } is the orbit of v under Γ, the sublattice L =
1 2 3 0
hv,v ,v ,v i ⊂ L is Γ-invariant. If µ ≥ 1, then U 6= ∅, and we see from
1 2 3 1
(2.4), (2.5) and (2.6) that rankL ≥ 3. If µ = 0, then U ,U ,U 6= ∅, and
0 1 2 3
again we see from (2.4), (2.5) and (2.6) that rankL ≥ 3. Thus rankL = 3
0 0
and L ≃J , whence by Proposition 1.6 L is not quasi-invertible.
0 Γ 0
Step 4. We show that L is a direct summand of L . Set m′ =|S|.
0 1
10 M. Borovoi and B. Kunyavski˘ı
First assume that µ ≥ 1. Choose u ∈ U ⊂ S. Set S′ = S r {u }. For
1 1 1
each s ∈ S′ (i.e., s 6= u ) consider the one-dimensional (i.e., of rank 1) lattice
1
X =he i. We obtain m′−1 Γ-invariant one-dimensional sublattices of L .
s s 1
Denote by Υ the set of pairs (ι,λ) such that 1 ≤ ι ≤ µ, 1 ≤ λ ≤ n , and if
ι
ι=1, then λ6=1,2. For each (ι,λ)∈Υ consider the one-dimensional lattice
Ξ =hε −ε i.
ι,λ ι,2λ−1 ι,2λ
We obtain −2+ µ n one-dimensional Γ-invariant sublattices of L .
ι=1 ι 1
We show that P
L =L ⊕ X ⊕ Ξ . (2.7)
1 0 s ι,λ
sM∈S′ (ι,Mλ)∈Υ
Set L′ =hL ,(X ) ,(Ξ ) i, then
1 0 s s6=u1 ι,λ (ι,λ)∈Υ
µ µ µ
rankL′ ≤3+(m′−1)−2+ n =m′+ (2n −1)− (n −1)=rankL .
1 ι ι ι 1
ι=1 ι=1 ι=1
X X X
(2.8)
Therefore, it suffices to check that L′ ⊃L . The set
1 1
{v}∪{e | s∈S}∪{ε −ε | 1≤ι≤µ,1≤λ≤n }
s ι,2λ−1 ι,2λ ι
is a set of generators of L . By construction v,v ,v ,v ∈ L ⊂ L′. We have
1 1 2 3 0 1
e ∈ X ⊂ L′ for s 6= u . By (2.4) e ∈ L′, hence e ∈ L′. By
s s 1 1 s∈U1 s 1 u1 1
construction
P
ε −ε ∈L′, for all (ι,λ)6=(1,1),(1,2).
ι,2λ−1 ι,2λ 1
From (2.6) and (2.5) we see that
µ µ
(ε −ε )∈L′, ξ′′ ∈L′ .
ι,1 ι,2 1 ι 1
ι=1 ι=1
X X
Thus
ε −ε ∈L′, ε −ε ∈L′ .
1,1 1,2 1 1,3 1,4 1
We concludethatL′ ⊃L ,henceL =L′. Fromdimensioncount(2.8)wesee
1 1 1 1
that (2.7) holds.
Now assume that µ = 0. Then for each κ = 1,2,3 we choose an element
uκ ∈Uκ andsetUκ′ =Uκr{uκ}. WesetS′ =U1′∪U2′∪U3′ =Sr{u1,u2,u3}.
Again for s ∈ S′ (i.e., s 6= u ,u ,u ) consider the one-dimensional lattice
1 2 3
X =he i. Weobtainm′−3one-dimensionalΓ-invariantsublatticesofL =L.
s s 1
We show that
L =L ⊕ X . (2.9)
1 0 s
s∈S′
M