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ЧЕБЫШЕВСКИЙ СБОРНИК
Том 20. Выпуск 2.
УДК 511.6 DOI 10.22405/2226-8383-2019-20-2-234-243
О некоторых характерах представлений групп
Д. Малинин
Малинин Дмитрий — кафедра математики и информатики, Университет Флоренции (Флоренция, Италия) e-mail: йтМгт,п@дт,ай.сот,
Аннотация
Мы изучаем поля реализации и целочисленность характеров дискретных и конечных подгрупп 6X2(0) и связанные с ним решетки, а также целочисленность характеров конечных групп G.
Теория характеров конечных и бесконечных групп играет центральную роль в теории групп, теории представлений конечных групп и ассоциативных алгебр. Классические результаты связаны с некоторыми арифметическими задачами: описание целочисленых представлений существенно для конечных групп над кольцами целых чисел в числовых полях, локальных полях или, в более общем случае, для дедекиндовых колец.
Существенная часть этой статьи посвящена следующему вопросу, восходящему к В. Бернсайду: каждое ли представление над числовым полем может быть сделано целочисленным.
Всякое ли линейное представление р : G ^ GLn(K) конечной группы G над числовым полем K/Q сопряжено в GLn(K) с представлением р : G ^ GLn(Ox) над кольцом целых чисел Ок толя К1 Чтобы изучить этот вопрос, используется связь целочисленых представлений и решеток.
Этот вопрос тесно связан с глобально неприводимыми представлениями; концепция, предложенная Дж. Томпсоном и Б. Гроссом, была изучена Фам Хыу Тиепом и обобщена Ф. Ван Ойстаеном и А. Е. Залесским, однако остается много открытых вопросов.
Нас интересуют арифметические аспекты целочисленной реализуемости представлений конечных групп, и, в частности, рассматриваются условия реализуемости в терминах символов Гильберта и алгебр кватернионов.
Ключевые слова: Гиперболические решетки, группы, порожденные отражениями, характеры дискретных и конечных групп, индекс Шура, дедекиндовы кольца, глобально неприводимые представления, простые алгебры над числовыми полями, кватернионы, решетки в простых алгебрах, символ Гильберта, роды, поля расщепления.
Библиография: 29 названий. Для цитирования:
D. Malinin. On some characters of group representations // Чебышевский сборник, 2019, т. 20, вып. 2, с. 234-243.
CHEBYSHEVSKII SBORNIK Vol. 20. No. 2.
UDC 511.6 DOI 10.22405/2226-8383-2019-20-2-234-243
On some characters of group representations
D. Malinin
Malinin Dmitry — Dipartimento di Matematica e Informatica U. Dini, Universita degli Studi di Firenze, (Firenze, Italy) e-mail: [email protected]
Abstract
We study realization fields and integrality of characters of discrete and finite subgroups of SL2(C) and related lattices with a focus on on the integrality of characters of finite groups G. Theory of characters of finite and infinite groups plays the central role in the group theory and the theory of representations of finite groups and associative algebras. The classical results are related to some arithmetic problems: the description of integral representations are essential for finite groups over rings of integers in number fields, local fields, or, more generally, for Dedekind rings. A substantial part of this paper is devoted to the following question, coming back to W. Burnside: whether every representation over a number field can be made integral. Given a linear representation p : G ^ GLn(K) of finite group G over a number field K/Q, is it conjugate in GLn(K) to a representation p : G ^ GLu(Ok) over the ring of integers Ok'- To study this question, it is possible to translate integrality into the setting of lattices.
This question is closely related to globally irreducible representations; the concept introduced by J. G. Thompson and B. Gross, was developed by Pham Huu Tiep and generalized by F. Van Oystaeyen and A.E. Zalesskii, and there are still many open questions. We are interested in the arithmetic aspects of the integral realizability of representations of finite groups, splitting fields, and, in particular, consider the conditions of realizability in the terms of Hilbert symbols and quaternion algebras.
Keywords: Hyperbolic lattices, groups generated by reflections, characters of discrete and finite groups, Schur index, Dedekind ring, globally irreducible representations, simple algebras over number fields, quaternions, lattices in simple algebras, Hilbert symbol, genera, splitting fields.
Bibliography: 29 titles. For citation:
D. Malinin, 2019, "On some characters of group representations" , Chebyshevskii sbornik, vol. 20, no. 2, pp. 234-243.
Dedicated to the 80th anniversary of the birth of Professor Michel Deza
1. Introduction
In this paper we are interested to study the integrality of characters of discrete subgroups of SL2(C) and related lattices.
Hyperbolic lattices in dimension three, that is, discrete cofinite subgroups of SL2(C), show a preference for having integrally valued character functions, see [29]. Probably the first known lattice with non-integral character seems to be the one presented by Vinberg at the very end
of his fundamental paper [29] where it plays the role of an example for reflection groups. We can present a version of this example and then discuss a series of lattices which contains, most probably, infinitely many with no integer valued character. This is a lattice in three-dimensional hyperbolic space generated by reflections. Let P be the solid in H3 described combinatoriallv as a prism with two opposite triangular and three planar quadrangular faces.
Proposition 1. (Vinberg [29]). The group r generated by reflections on the faces of P is a cofinite but not cocompact lattice in hyperbolic space H3. It is not arithmetic.
Consider the following presentation of a subgroup ri of r
Generators: ai,a2,ri,r2,
Relations:
(1) af = af = (am)2 = (a2n)2 = (afTf)2 = (t2-1ti)2 = I,
(2) (aiTf)3 = /,
(3) rf = rf = I.
Proposition 2. r1 has trace field equal to Q(\/-3), the field of cube roots of unity. Its character values (squared) are unbounded at the nonarchimedian valuation at the prime 2 and integral at all other non-archimedian places. It is cofinite with exactly one cusp.
The character of r1 is determined by the following representation G ^ GLn(C)
ai =( - ^l)^2 =(i 0),
_ ( \ _( ^3+ o \
Ti = ^ -V2i ,T2 = ^ 0 )
In [10] H. Helling considered explicit hyperbolic manifolds obtained by Dehn surgeries of type (An, n) on the figure of eight knot 4i. These share the properties of an earlier paper [15] and the above propositions of having associated lattices SL2(C) with non-integrallv valued character functions. See also [15] and [27]. This gives a series of examples of lattices SL2(C) having nonintegral characters.
2. Integrality of characters for finite groups
Starting from this section in this paper we focus on the integrality of characters of finite groups G. Though the traces of g e G are always algebraic integers, the representations G ^ GLn(K) are not always realizable in the rings of integers of algebraic number fields K.
Let us consider the following
Assumption 1. Let G be a finite group, K a number field with the ring of integers Ok and p : G ^ GLn(K) an irreducible representation of G. We denote by Mk the associated irreducible KG-module.
Definition. The representation p : G ^ GLn(K) is integral, if and only if p(g) e GLu(Ok) for all g e G. We say that p(G) can be made integral, if and only if there exists an integral representation G ^ GLu(Ok) which is equivalent to p. We call Mk integral if p(G) can be made integral.
In other words, p(G) can be made integral if and only if we can apply a base change such that all matrices have integral entries.
W. Burnside asked the question whether every representation over a number field can be made integral. To study this question, it is possible to translate integrality into the setting of lattices.
Question. (W. Burnside, I. Schur, later W. Fe.it,, J.-P. Serre). Given a linear representation p : G ^ GLn(K) of finite group G over a number field K/Q, is it conjugate to a representation p : G ^ GLn(Ox) over the ring of integers Ok?
There is an algorithm which efficiently answers this question, it decides whether this representation can be made integral, and, if this is the case, a conjugate integral representation can be computed. Integral realizability of p ver the ring of integers Ok depends strongly on the class number cIk of K. The following proposition is well-known, see e.g. [4].
Proposition 3A. Assume that one of the conditions hold:
(i) We have K = Q.
(ii) We have cIk = 1-
(Hi) We have the greatest common divisor GCD(cIk; n) = 1.
(iv) We have cIk/cl2K = 1.
Then the representation p : G ^ GLn(K) can be made integral.
In the papers by D. K. Faddeev (1965, 1995), see [6] and [7], some new ideas on generalized integral representations over Dedekind rings were discussed.
The following theorem is contained in [2].
Theorem 1 (Cliff, Ritter, Weiss, [2]). Let G be a finite solvable group. Then every absolutely irreducible character x of G can be realized over Z[(m], where m is the exponent of G.
Example. The metacvclic group G = (x; y\x9 = y19 = 1; yx = y7) admits an absolutely irreducible representation G ^ GL3(K) which cannot be made integral, where K is the unique subfield of Q((57) of degree 12.
Theorem 2 (Serre, [28]).
Let G = Qs, K = Q(V-d), and d> 0. Then
1) G is realizable over K, p : G ^ GL2(K) if and only if d = a2 + b2 + c2 for some integers a, b, c.
2) G is realizable over Ok, p : G ^ GL2(Ok), if and only if d = a2 + b2 for some integers a, b or d = a2 + 262 for some integers a, b.
The starting point of studying absolutely irreducible representations of finite groups with the property of irreducibilitv modulo all primes was the concept of of global irreducibilitv. The notion of globally irreducible representations for the ring of rational integers appeared in papers by B. H. Gross, see [8], [9] in order to explain new series of Euclidean lattices discovered by N. Elkies and T. Shioda using Mordell-Weil lattices of elliptic curves.
The concept of global irreducibilitv for arithmetic rings has been introduced by F. Van Ovstaeven and A.E. Zalesskii: a finite group G c GLn(F) over an algebraic number field F is globally irreducible if for every non-archimedean valuation v of F a Brauer reduction reduction of G (mod v) is absolutely irreducible. The following theorem is proven in [25].
Theorem 3 (F. Van Ovstaeven and A.E. Zalesskii, see [25]).
O.F-span OfG of a group G c GLu(Of) is equal to Mu(Of) if and only if G c GLu(Of) is globally irreducible.
The natural problem is to describe the possible n and arithmetic rings Of such that there is a globally irreducible G c GLu(Of)■ In our particular situation it is interesting, what happens for n = 2? This question was considered in [20], [22]. The answer is given in the theorem below, see [22], Theorem, p. 9.
Theorem 4 ([22]).
1) Let G = Q4m be the group of generalized quaternions, and let H = G = Qs be the group of quaternions. Then there is a quadratic subheld Ki C K and an Okx H-module I which is an ideal in an extended held Li = Ki(i), such that: G = Q4m is realizable over Ok if and only if H is realizable over O^, and all Hilbert svmbols ^ = i for an
2)IfG = Q4m is not realizable over Ok, the minimal realization held such that H is realizable over its ring of integers is a biquadratic extension Q(v/dT, vd), where d = did2 an d di, d2 are integers not equal to ± 1 or to ± d.
3) The explicit computation of I in Li = Ki(i) is relevant to a representation of the integer d = a2 + b2 + c . jKx of either of these ideals is a principal ideal in Okx if
(1) b = c; then d = a2 + 2b2 ((a, b) = 1) - equivalently, d has no prime factors p = 5(mod8) and p = 7(mod8), or
(2) c = 0; then d = a2 + b2 ((a, b) = 1) or equivalently, d has no prime factors p = 3(mod4).
Let G be a finite group and % its complex irreducible character. A number field K/Q is a splitting field of %, if there exists a representation of G over K affording %. A splitting field K is of the minimal degree, if there is no splitting field of % with degree smaller than K. We say that a splitting field K of % is integral, if any representation of G over K affording % can be made integral.
Otherwise, the splitting field K is nonintegral.
%%
have the same relative degree over the character field Q(%), which is called the Schur index of % over Q, [18] . Let us use for this degree the following notation: ^q{x)(%)-
For each place v of Q(%), there is an associated local Schur index of % at v, denoted by ^q(x)v (%), and the least common multiple
mQ(x)(%) = LCMV {mQ(x)v (%M
The field Q(%) C K is a splitting field of % if and only if Wq(x)v(%) divides [Kw : Q(%)v] for all places v of Q(%) ^^d all divisors w of v.
If m,Q(%) > 1, there ^re infinitely many minimal splitting fields of % and if wq(%) = 1, then the field of characters Q(%) is the unique minimal splitting field of %.
Do there exist integral and nonintegral minimal splitting fields of a given character? If so, how many are there?
Let us consider the case of trivial Schur index. In this case Q(%) is the only minimal splitting %
% with Q(%) = Q the minimal splitting field of % is integral. Thus in general both cases will occur. We will now concentrate on the case wq(%) > 1, more precisely on the case wq(%) > 1, Q(%) = Q and deg(%) = 2.
Let G be a finite group, K a number field with the ring of integers O = Ok- We will now concentrate on a special situation, originally treated by Serre in [Ser08], for which the existence of integral and nonintegral minimal splitting fields is closely connected to the theory of quaternion algebras and Hilbert symbols. Below we consider (the Hilbert symbol (a, b) over Q and for a place
v of Q we denote by (a; b)v the corresponding local Hilbert svmbol over Qv. By Br2(Q) we denote
Q
We denote by GIk the group of ideal classes of K. For a finitely generated O^-module (a lattice) M we denote by cl(M) its Steinitz class. The simple component of QG, corresponding to the irreducible character %, is a non-split quaternion algebra over Q, which we denote by D. The proof of the proposition 3 below is contained in the paper by J.-P. Serre [28].
Definition (see [28].) Let K be an imaginary quadratic number field with discriminant —d,d> 0. We define the map
: CIK/CI2K ^ B^(Q); [a] ^ (N(a), —d).
Proposition 3 (see [28]). Let K be an imaginary quadratic number Geld with discriminant —d, which splits D and which we consequently view as a subfield of D. Then the following conditions hold true:
(i) The map ex is well-defined and injective.
(ii) Let R be a maximal order of D containing O. Then the O-module R is G-invariant. In particular R is an OG-lattice.
(Hi) If R and R0 are two maximal orders of D containing O, then cl(R) = cl(Ro) in CIk/Cl2K. (iv) Let R be a maximal order of D containing O. Then we have ex(cl(R)) = (D) ■ (do; —d), where dp is the product of all primes ramified in D including —1 if to is ramified, (D) is the class of D B 2(Q)
Proposition 4 ([22], proposition 5).
(1) An algebraic number field K is a splitting field for the group G of quaternions if and only if K is totally imaginary and for all localizations Kv for all prime divisors v of 2 the local degree [Kv : Q2] is even.
(2) If K is a splitting field for the group G of quaternions, then [K : Q] is even.
(3) K is a splitting field for the group G of quaternions and K/Q is abelian, then K has a quadratic subfield Q(\/d).
For the convenience of the reader we include the proof of proposition 4.
Proof. By the theorem of Hasse-Brauer-Noether, K is a splitting field for (G)q if and only if the localization Kv is a splitting field locally for (G)qp = QpG for all prime divisors v of p. Since the quaternion algebra has invariants 1/2 at 2 and to in the Brauer group, and 0 at all other primes p, K is a splitting field for G if and only if K is totally imaginary and for all localizations Kv for all prime divisors v of 2 the local degree [Kv : Q2] is even [5], Satz 2, ch. VII, sect. 5. Since [K : Q] is the sum of [Kv : Q2], it must be even, and this implies (2).
If K/Q is abelian, its degree is even, and its Galois group has a subgroup of index 2, therefore,
Q
This completes the proof of proposition 4. Let us consider the following
Assumption 2. Let G be a finite group and let % be an irreducible character of a finite group with mQ(x) > 1, Q(x) = Q and deg(x) = 2.
Consider the simple component D of QG, corresponding to the irreducible character which was used in proposition 3 above. Below we consider classes of sublattices L(R) of a maximal order R of D. Recall that a quaternion algebra is just a 4-dimensional Q-algebra with center Q. We have the following equivalence:
(1) A quadratic field K is a splitting field of
(2) all places v of Q with mQv(x) = 2 do not split in K over Q,
(3) the field K can be embedded as a maximal subfield of D,
(4) For all places v of Q at which D is ramified, the field Kw splits ^ for all places w of K
Let K be an imaginary quadratic field which splits Then we can view D as a KG-module, which we denote by M^, and we have seen that a maximal order R of D containing O is an OG-lattice of Mk- To determine integrality, it is now sufficient to consider the set cl(L(R)) of classes of sublattices of R or, since ex is injective, the set ex(cl(L(R))). Let K be a minimal splitting field of the character %. Let us denote by Sk the set of prime ideals of O = Ok such that a Brauer reduction of Mk is reducible, let S' = S'K be the set of rational primes lying over ideals in Sk- Let S be the intersection of all S' = S'K over all minimal splitting fields K\ following [28], we denote by e(D, K) = eK(cl(R)) fa a maximal order R of D containing O. Remind that —d is the discriminant of K.
Lemma 1.
eK(cl(L(MK))) C e(D,K) ■ {npeSo(p, — d)ISo C 5}
Proof. It follows from [26], theorem 2.5, and the observation that the class of a sublattice of R can only change by a square or the class of [I] e CIk/Cl2K, where I is a prime ideal whose /-reduction is reducible, that cI(L(Mk)) C cl(R) ■ {niPes0 [I]|50 C S}. By applying the map eK to the equation obtained, we get eK(cI(L(Mk))) C e(D,K) ■ {npeso(pf(I), —d)|50 C S}, where f (I) is the inertia index of f (I) in K/Q. Assume t hat I e Sk, but not in S and ide al of K
above p. Then there exists a minimal splitting field L and a prime ideal q of L lying above p such that the reduction of Mk modulo I is reducible, while the reduction of modulo q is irreducible.
II norm N (I) = p2 and therefore (p?(I), —d) = 1.
This completes the proof of lemma 1.
Lemma 2 ([,3], Theorem 5.3.2, see also [24], Theorem 2.8, compare also [16], sect 81, p. 144, Theorem 112). Let (di)i£i he a finite set of elements of Q*, and let (ei,v)iej,vep be a set of numbers equal to ±1. There exist an infinite nu mber of x e Q* such th at (di,x) = €i,v for a Hi e I and all v e P if and only if the following three conditions are satisfied:
(1) Almost all of the €i,v are equal to 1, say, €itV = 1 for v / Po and a finite subset P0 C P.
(2) For all i e I we have (ei,v) = 1.
(3) For all v e P there exists xv e QV such th at (a,i, xv )v = €i,v for a 11 i e I.
Note that infiniteness of the number of x follows from Dirichlet's theorem on primes in arithmetic progressions which is involved in the proof.
Lemma 3. There is an infinite number of splitting fields K = Q(\/—d) of % such that CIk/ClK = 1.
Proof. It follows from [12] that CIk/Gl2K = 1 for quadratic fields K = Q(V—p) for —p = l(mod4). Let T = U^ be the set of rational primes such that Schur indices of % at Qi are 2. An extension K of the character field Q is a splitting field of if all places of K aSove the p e T inertia degrees divisible by 2. The Legendre symbol (^p) = (—l)(qi-i^/2{^^ for the disrnm^nt dK = —p. It follows from proposition 4 that for primes qi = 2 the character % splits iff = (—l)(?i+i)/2. For qi = 2 we can see that the inertia degree is 2 iff = — p = l(mod8). Now we can use Dirichlet's theorem on primes in arithmetic progression to conclude that there are infinitely many primes p satisfying the above congruence conditions for all Pi. This completes the proof of lemma 3.
Theorem 5. Let % be an irreducible character of a finite group with m,Q(%) > 1, Q(%) = Q
and deg(%) = 2. Then there exist infinitely many integral minimal splitting fields of %, and there is
%
Proof.
1) We can use proposition 3, (iv) together with lemma 3 to prove that there exist infinitely many integral minimal splitting fields of It follows from proposition 4 that the infinite number of K from lemma 3 are minimal splitting fields of
2) Let P be the set of all finite rational primes and to. Let Ram(D) be the set of all finite ramified primes in ^together w ith —1 in the case if D is ramified at to. Let for the elements Pi £ S U Ram(D) the set ei = ±1 be prescribed elements. It follows from [11], ch 5, sect. 6, and Dirichlet's theorem on primes in arithmetic progressions that there is an infinite number of primes q and integers x such that Hilbert symbols (pi,x)q = e» for all indices i.
According to proposition 3, (iv) and lemma 1, it is sufficient to prove that the unit class is not contained in the set of classes ( D) ■ {n^so (P, — rf) i^o c S} fa any S0 c S1 = S U Ram(D) and for an infinite number of d\ D is a non-split algebra, and we can assume that S0 is not empty. For any So c S U Ram(D) take dements 6p ± 1,p £ such that n^gi €p — —1; according to the above argument there are integers x and a prime q1 £ S1 such that D splits at q1 and (p,x)qi = ep for all p £ 51, thus ((D) ■ npeso (p, —x))qi = npeso (p,x)qi = —1 since (p, —1)qi = 1. Also we can take q1 = q'( if the corresponding S1 = S'{. Now we can use lemma 2 for I = S1, {ai}iei = [p}pes1, £i,v = £p, P0 = {q1}, where q1 corresponds to S0 c S.
The sufficient conditions for an imaginary quadratic field K = Q(\/—d), where d > 0, to be a splitting field of D is that for all q £ Ram,(D) the condition (q,d)q = (—1)(^+1)/2 if q = 2, or the condition (q,d)2 = 1 if q = 2 hold true; since K is imaginary, the sufficient condition at the infinite place is also satisfied. We can also assume that since the conditions for K = Q(V—d) to be a splitting field affect only v £ Ram(D) which do not intersect P0, and the second claim of theorem 5 follows from lemma 2.
Remark 1. A similar theorem holds in a more general settings, we have can find minimal integral and nonintegral splitting fields for a large number of characters of various groups assuming that x is an irreducible character of G with mQ(x) > 1-
Remark 2. In some earlier papers, see e.g [21], the author considered the conditions of integrality for representations of finite groups together with conditions of stability of Galois action. The following question has a deep topological motivation, see [1].
Let p : G ^ GLn(C) be a complex n-dimensional representation of a finite group G. Let t be an automorphism of the field C, not necessarily continuous. For g £ G, we act by t on the matrix coefficients of p(g) and obtain a new matrix r(p(g)).
We obtain a new subgroup r(p(G)) in GLn(C). Is it possible that the subgroup r(p(G)) is not conjugate to p(G) in GLn(C), i.e. there is no matrix X £ GLn(C) such that r(p(G)) = Xp(G)X-1?
Acknowledgement. The author is grateful to the referee for useful remarks and suggestions.
3. Conclusion
Theory of characters of finite and infinite groups plays the central role in the group theory and the theory of representations of finite groups and associative algebras. The classical results are related to some arithmetic problems: the description of integral representations are essential for finite groups over rings of integers in number fields, local fields, or, more generally, for Dedekind rings. In this paper we are interested to study the integrality of characters of discrete and finite subgroups of SL2(C) and related lattices. A substantial part of this paper is devoted to the following question, coming back to WT. Burnside: whether every representation over a number field can be made integral. To study this question, it is possible to translate integrality into the setting of lattices.
Question. (W. Burnside, I. Schur, later W. Fe.it,, J.-P. Serre). Given a linear representation p : G ^ GLn(K) of finite group G over a number field K/Q, is it, conjugate to a representation p : G ^ GLn(Ox) over the ring of integers Ok?
This question is closely related to globally irreducible representations; the concept introduced by J. G. Thompson and B. Gross, was developed and generalized by Pham Huu Tiep, F. Van Ovstaeven and A.E. Zalesskii, and there are still many open questions. We are interested in the arithmetic aspects of the integral realizability of representations of finite groups, and, in particular, prove the existence of infinite number of splitting fields where the representations are not realizable.
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