View of Galois embedding problems and symmetric powers

GALOIS EMBEDDING PROBLEMS AND SYMMETRIC POWERS

TERESA CRESPO and ZBIGNIEW HAJTO^∗

Abstract

We consider a Galois embedding problem given by a finite projective group and its special linear lifting. We give an equivalent condition to the solvability of such an embedding problem and a method of construction of its solutions.

LetGbe a finite (subgroup of the) special linear group of degreenover a field Kof characteristic 0 and letGbe the projectivized group ofG. The groupG is then a group extension ofGby a cyclic group of orderpdividingn. Given a Galois realizationL|Kof the groupG, we consider the Galois embedding problem

(GEP) G→GGal(L|K).

In this paper, we give a correspondence between the proper solutions to (GEP) and theK-defined points of a certain constructible set in aK-vector space. We present an explicit method to obtain a proper solution to the con- sidered embedding problem from such a point. These results are a generaliza- tion of the methods obtained in [2] and [3].

Let ρ:G→SL(n, K)

be the faithful special linear representation of the groupG. AsGis the image of Gunder the projection of the linear group GL(n, K)onto the projective group PGL(n, K), the kernel ofG→→Gis the subgroup of the homotheties of ratio apth root of unity, for somepdividingn. This implies that thepth symmetric powerρ^(p)ofρfactors through the groupGand gives a representation

ρ:G→SL(m, K), wherem=_n+p−1

p

.

∗Both authors are partially supported by grant BFM2003-01898, Spanish Ministry of Educa- tion.

Received May 28, 2004.

Assume (GEP) is solvable and letLbe a proper solution. We considerLas aK-vector space and the representation

φ:G→GL(L)

given by the Galois action. By the normal basis theorem,φis the regular rep- resentation and soLcontains an invariantK-vector spaceU = u1,· · ·, un of dimensionnsuch thatφ|Uis equivalent to the faithful unimodular represent- ationρofG. LetU^(p)denote the vector space of thepth symmetric power of the representationφ|Uand consider inU^(p)the basis(u⁽ⁱ1¹⁾·. . .·u⁽ⁱ_nⁿ⁾)i1+···+in=p. We obtain a morphism ofK[G]-modulesU^(p)→Lgiven byu⁽ⁱ1¹⁾·. . .·u⁽ⁱ_nⁿ⁾→ uⁱ1¹·. . .·uⁱ_nⁿ. MoreoverL=L(u1), due to the fact that a generator of the group Gal(L|L)Ker(G→→G)sendsu¹toζu¹, forζa primitivepth root of unity.

Now letL|Kbe a Galois extension realizingGand consider the represent- ation

G→GL(L)

given by the Galois action. Again by the normal basis theorem, this repres- entation is the regular one. Let us assume that the representation ρ = ρ^(p) is contained in the regular representation ofG, i.e. that for each irreducible representation contained inρthe number of times it is contained is not greater than its dimension. We shall then determine all possible copies ofρcontained in the regular representation ofG.

Examples. We now give examples of groupsGsatisfying the conditions above, different from the ones considered in [2] and [3].

1) The groupsH216^SL3,H72^SL3,F36^SL3are primitive unimodular groups of degree 3 (see [4] or [5]). They are triple covers of the corresponding projectivized groups H216, H72, F36. We observe that the representations of H216^SL3 and H72^SL3 are defined over the fieldQ(ζ⁹), whereζ⁹denotes a primitive ninth root of unity, and the representation ofF36^SL3is defined over the fieldQ(ζ3), whereζ3denotes a primitive third root of unity. The decomposition in a sum of irreducible representations of the third symmetric power of these representations is given in [6], Table 2. It shows that this third symmetric power is in each case contained in the regular representation of the corresponding projectivized group.

2) The non trivial triple cover 3A7of the alternating groupA7has a faith- ful special linear irreducible representation of dimension 6, which is defined over the field Q(ζ3), where ζ3 denotes a primitive third root of unity. As a

dimension 6 special linear group, 3A7can be generated by the matrices









1 0 0 0 0 0

0 ζ3² 0 0 0 0

0 0 1 0 0 0

0 0 0 0 0 1

−1+ζ3 0 1−ζ3 −ζ3 ζ3 1

2 0 −1 −1 0 −1







,









0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

1 0 0 0 0 0

−1 1 −ζ3 0 ζ3 1







,

mapping respectively to the permutations (123) and (34567) ofA7(see [7]).

By computation we obtain that the third symmetric power of this represent- ation decomposes in the sum of five irreducible representations of dimensions 1, 6, 14, 14, 21. Namely, the corresponding character isχ1+χ2+χ5+χ6+χ8

with the notation in [1].

We shall use the following lemma on representations which is a straight- forward generalization of Lemma 1 in [3].

Lemma 1. Let Vk be K-vector spaces of dimensionnk and ϕk : G → GL(Vk)non equivalent absolutely irreducible representations,k = 1, . . . , l. We consider:

φ=ϕ1^r1⊕ · · · ⊕ϕ^r_l^l :G→GL(V ) φ=ϕ1^s1⊕ · · · ⊕ϕ^s_l^l :G→GL(V)

whereV =V1^r1⊕· · ·⊕V_l^rl,V =V1^s1⊕· · ·⊕V_l^sl, withsk ≤rk,k=1, . . . , l. Let us fix monomorphismsfk,j :Vk →V_k^rk such thatπj ◦fk,j :Vk →Vk, where πj is the projection on thej-component, is an isomorphism of G-modules, j =1, . . . , rk,k=1, . . . , l.

Then every invariantK-subspace ofV isomorphic toV as aG-module is a direct sum of invariantK-subspaces isomorphic to eachV_k^sk and each of these direct summands has a basis of the form

rk

j=1

a_j^pfk,j(v^k_i)

1≤i≤nk,1≤p≤sk

for some a_j^p ∈ K, (v1^k, . . . , v_n^k_k) a K-basis of Vk, k = 1, . . . , l and rank(a_j^p)1≤j≤rk,1≤p≤sk =sk.

We now state our main result.

Theorem1. LetGbe a finite subgroup of the special linear group of degree nover a fieldKof characteristic0and letGbe the projectivized group ofG. Let pbe the order of the kernel ofG→Gandm=_n+p−1

p

. Letρbe the faithful

special linear representation of the groupGandρ=ρ^(p). Letρ=_l

k=1ρ_k^sk, withρknon equivalent irreducible representations of dimensiondk. We assume sk ≤dk, fork=1, . . . , l. LetL|Kbe a Galois realization of the groupG.

There exists a constructible setQinK^m such that the Galois embedding problem

(GEP) G→GGal(L|K)

has a proper solution if and only ifQhas a point defined overK.

From a point inQ(K)we can construct an elementγ in Lsuch that the extensionL=L(^p√γ )is a proper solution to(GEP).

Proof. We consider the representation G→GL(L)

given by the Galois action. As stated previously, this representation is the regular one and so containsdk times each representationρk. For eachk, let Vk,1, . . . , Vk,dk bedkK-subspaces ofL, in direct sum, invariant by the Galois action and such that the restriction of the Galois action to eachVk,jgives the rep- resentationρk. Let us choose a basis(v_ij^k)1≤i≤dkin eachVk,jsuch thatv^k_ij →v_ij^k defines a morphism ofG-modules fromVk,j toVk,j. By applying the lemma, we find that eachG-submoduleWofLsuch that the restriction of the Galois ac- tion to it gives the representationρhas a basis(wipk)¹≤i≤nk,1≤p≤sk,1≤k≤l, where wipk=_rk

j=1a_pj^k v_ij^k for somea_pj^k inKsatisfying rank(a_pj^k )1≤j≤rk,1≤p≤sk =sk, fork =1, . . . , l.

Now letU be the space of the representationρand letρbe given in a basis (ui)1≤i≤n. LetU^(p)denote the vector space of thepth symmetric power of the representationρand consider inU^(p)a basis(vi¹,...,in)i¹+···+in=p such that the linear map given by

vi1,...,in →u⁽ⁱ1¹⁾·. . .·u⁽ⁱ_nⁿ⁾

is a morphism ofG-modules. LetF be an isomorphism ofG-modules from U^(p)toW. The vector spaceWis a symmetric power if the elementsF (vi1,...,in) can be written asuⁱ1¹uⁱ2²· · ·uⁱ_nⁿ for some elements (ui)1≤i≤n in an algebraic extension of the fieldL.

We now consider the map

(1) Lⁿ→L^m, (ui)→(uⁱ1¹uⁱ2²· · ·uⁱ_nⁿ),

whereLdenotes an algebraic closure of the fieldL. We want to see that the intersection ofL^mwith the image of this map is an algebraic set inL^m. We consider all equations of the form

(2) vi1,...,invi1,...,in −vj1,...,jnvj1,...,jn =0

withik+i_k =jk+j_kfork=1, . . . , n. It is clear that the vectors(uⁱ1¹uⁱ2²· · ·uⁱ_nⁿ) satisfy all these equations. Conversely, if(vi¹,...,in) ∈L^msatisfy all the equa- tions, then we can takeu1 in a finite extension ofLsuch thatvp,0,...,0 = u^p1

andu2, . . . , unsuch that uk

u1 = vp−1,0,...,1,0,...,0

vp,0,...,0

where the subindex 1 of the element in the numerator is in the placek. Then (u1, . . . , un)maps to(vi1,...,in)under (1).

Now the vector spaceW is a symmetric power if the elementsF (vi1,...,in) satisfy all equations (2). We then obtain a family of algebraic equations in the elementsa_pj^k with coefficients inLdefining an algebraic varietyA. Now, the condition that the elementsF (vi1,...,in)be of the formuⁱ1¹uⁱ2²· · ·uⁱ_nⁿ is invariant by the action of the groupGand soAis defined overK. LetQbe the subset of Adefined by the equalities rank(a_pj^k )¹≤j≤rk,1≤p≤sk =sk. For(a_pj^k )inQ(K), the fieldL=L(√^p γ ), withγ =F (vp,0,...,0)is a cyclic extension ofLcontaining the elements^p

F (vp,0,...,0), . . . , ^p

F (v0,...,0,p)which are a basis of aK-vector subspace ofLon which the action ofGcorresponds to the representationρ. Then [L(√^p γ ):L]=pand we obtain the statement in the theorem.

Remark. If the fieldKis a differential field, the elements ^p

F (vp,0,...,0), . . . ,^p

F (v⁰,...,0,p)in the proof of the theorem, constructed from(a_pj^k )∈Q(K), are a basis of the space of solutions of a homogeneous linear differential equa- tion with Galois groupG.

REFERENCES

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2. Crespo, T., Hajto, Z.,Differential Galois realization of double covers, Ann. Inst. Fourier (Grenoble) 52 (2002), 1017–1025.

3. Crespo, T., Hajto, Z.,The Valentiner group as Galois group, Proc. Amer. Math. Soc. 133 (2005), 51–56.

4. Miller, G. A., Blichfeldt, H. F., Dickson, L. E.,Theory and Applications of Finite Groups, John Wiley and Sons, Inc., 1916.

5. Singer, M. F., Ulmer, F.,Liouvillian and algebraic solutions of second and third order linear differential equations, J. Symbolic Comput. 16 (1993), 37–73.

6. Singer, M. F., Ulmer, F.,Galois groups of second and third order linear differential equations, J. Symbolic Comput. 16 (1993), 9–36.

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DEPARTAMENT D’ÀLGEBRA I GEOMETRIA UNIVERSITAT DE BARCELONA

GRAN VIA DE LES CORTS CATALANES 585 08007 BARCELONA

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E-mail:teresa.crespo@ub.edu

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