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Numerical Groups 3 NUMERICAL GROUPS MAR´IA TERESA LOZANO AND JOSE´ MAR´IA MONTESINOS-AMILIBIA Dedicated to our friend Elena Mart´ın Peinador on the occasion of her 70th birthday Abstract. A group of matrices G with entries in a number field K is defined to be numerical if G has a finite index subgroup of matrices whose entries are algebraic integers. It is shown that an irreducible or completely reducible subgroup of GL(n,K) GL(n, C) is numerical if and only if the traces of its elements are⊂ algebraic integers. Some examples are given. 1. Introduction Speaking loosely, two properties characterize an arithmetic group of matrices with entries in a number field K. One is that it has a finite index subgroup of matrices whose entries are algebraic integers (we call such groups numerical). The second property is related to the number of places of the field K (see [12], [5] and [2]) and will not be considered here. The purpose of this paper is to analyze the definition of numerical group and characterize it in terms of traces. This will be very useful when ascertaining if a given group of is arithmetic or not. The paper is organized as follows. In Section 2 we recall some known concepts on complex matrix subgroups of GL(n, C). We prove for them that the condition of being irreducible or completely irreducible is characterized by the trace form (Theorem 2.6). arXiv:1911.11435v1 [math.GR] 26 Nov 2019 In Section 3 we define the concept of numerical in a general frame- work, for subgroups of automorphisms of a vector space V over C. The main results are the following theorems Theorem 3.2. Let K be a number field and let G be a irreducible or completely reducible subgroup of GL(n, K) GL(n, C). Then G is numerical if and only if the trace of all the elements⊂ of G are algebraic integers. Moreover, G is real numerical if K is real. Date: November 27, 2019. 2000 Mathematics Subject Classification. Primary 11E57, 11H56, 20H20. Key words and phrases. numerical group, matrix group, trace of matrices. Partially supported by grant MTM2016-76868-C2-2-P. 1 2 M.T.LOZANOANDJ.M.MONTESINOS-AMILIBIA Theorem 3.1. Let G be a group of automorphisms of a vector space V over C of dimension n. Then G is a numerical group if and only if there is a representation ρb(G) such that ρb(G) GL(n, K )), where ⊂ O K is the ring of algebraic integers in a number field K, which is real ifO G is a real numerical group. The proof of this Theorem is constructive and we will use this in a forthcoming paper to revisit [5] and obtain the list of all the arithmetic orbifols with the Borromean rings as the singular set in a new way. In the last section, Section 4, we give some examples of numerical and no numerical subgroups in a family of subgroups of SL(2, C) related to Hecke and Picard modular groups. 2. The trace form on irreducible and completely irreducible subgroups of GL(n, C) Le K be a field. Each element of the K-vector space V = Kn is a (n 1)-matrix whose entries are in K. Let× M(n, K) be the K-vector space of (n n)-matrices with entries in K, and let GL(n, K) be subset of invertible× matrices (it has a group structure under the matrix multiplication). There exits a left action of M(n, K) on V by matrix multiplication. These actions are the en- domorphisms of V . The actions of the elements of GL(n, K) are the automorphisms of V . Two matrices A, B M(n, K) are similar if there exists a matrix U GL(n, K) such that∈ U −1AU = B. Because similar matrices repre- sent∈ the same endomorphism with respect to (possibly) different bases, similar matrices share all properties of the endomorphism that they rep- resent, for instance, the invariants derived from the characteristic poly- nomial, as determinant and trace. Two groups G,H GL(n, K) are equivalent if there exits a matrix U GL(n, K) such that∈ U −1GU = H. We can also say that two groups of∈ endomorphism of V are equivalent if they are represented by equivalent subgroups in GL(n, K). (They are related by a change of basis and therefore they share all algebraic properties as linear groups.) The equivalence is used to work in a more convenient framework. Given a subset S M(n, K), a subspace W V is call S- invariant space if A(W ) W⊂for all A S. A S-invariant⊂ space W is minimal if 0 and W are⊂ the only subspaces∈ of W which are S-invariant. {A} subgroup G of GL(n, K) is called irreducible if V and 0 are the only minimal G-invariant subspaces. A subgroup G of GL{(n,} K) is called completely reducible if there is a direct sum decomposition NUMERICAL GROUPS 3 r V = Vi such that the restriction G V is irreducible, for each i ⊕i=1 | i ∈ 1, ..., r . In this case each Vi is a minimal G-invariant subspace. { Let G} be a subgroup of GL(n, K), W V a minimal G-invariant space of dimension w and b is a basis of W ,⊂ the action of G on b defines a homomorphism ρ : G ρ(G) GL(w,K). −→ ⊂ This map ρ is an irreducible representation of G of degree w. Two repre- sentations ρ and ρ′ are equivalent if there exits a matrix U GL(w,K) such that U −1ρ(A)U = ρ′(A) for each A G, both representations∈ are related by a change of basis. ∈ Let C denote the complex field. Let G be a completely reducible subgroup of GL(n, C). Then Cn is a direct sum of minimal G-invariant spaces. These subspaces can be ordered such that the actions of G on the first k1 are equivalent; the actions of G on the next k2 are also equivalent, and so on. There exists a matrix U GL(n, C) such that for each A G the matrix U −1AU is like this ∈ ∈ k1 k1 k1 Diagonal[A1, ...A1; ...; Ai, ...Ai; ...; As, ...As] where ρi : A Ai is anz irreducible}| { z representation}| { z }| of{ degree, say, ni; −→ n = n1k1 + ... + nsks; ki is the multiplicity of ρi and for i = j the 6 representations ρi and ρj are nonequivalent. In this way we obtain a new basis in Cn and a group U −1GU equivalent to G. The linear hull C[G] of G is defined as the set of all linear combina- tions ( i λigi), λi C, gi G. It is the vector subspace on M(n, C) generated by the matrices∈ of∈ G. C[G] is also a ring and an associative algebraP over C with unity I, where the product is the multiplication of matrices λigi λjgj = λiλjgigj i ! j ! i,j X X X The following theorem is a generalization of the Artin-Wedderburn Theorem (See [11, Section 14.4]). Theorem 2.1. Let G GL(n, C) be a completely reducible group. ⊂ There exist integer positive numbers n1, .., ns,k1, ..., ks and a matrix U GL(n, C) such that U −1C[G]U consists of all matrices of the fol- lowing⊂ form Diagonal[A1, ..., A1; ...; Aj, ..., Aj; ...; As, ..., As] where the multiplicity of Ai is ki and A1, ..., As 4 M.T.LOZANOANDJ.M.MONTESINOS-AMILIBIA take independently all possible values in M(n1, C), ..., M(ns, C) and n = n1k1 + ... + nsks. In particular, if G is irreducible then C[G]= M(n, C). Definition 2.1. The trace form T is the bilineal symmetric form de- fined by T : M(n, C) M(n, C) C (A, B×) −→ T (A, B)= tr(AB) → where tr(A) denotes the trace of the matrix A. If G GL(n, C) is a ⊂ group, the restriction of the trace form to C[G] is denoted by TG. The main result in this paragraph (Theorem 2.6) is the characteri- zation of an irreducible or completely reducible group G GL(n, C) ⊂ by means of the trace form TG. Claim 2.1. The trace T is a nondegenerate symmetric bilinear form on M(n, C), that is, if tr(AB)=0 for all B M(n, C) then A =0. ∈ Proof. Let A =(aij) be a matrix in M(n, C) such that tr(AB)=0 for all B M(n, C). Denote by Eij the matrix with all entries 0 but the ∈ 2 entry in position ij which is 1. The n matrices Eij form a basis of the vector space M(n, C). Therefore tr(AEij) = aij = 0 for all i, j, this implies A = 0. An analogous demonstration will be used to prove the following re- sult, which is the first part of the looked for characterization. Theorem 2.2. Let G be an irreducible or completely reducible subgroup of GL(n, C). Then TG is a nondegenerate symmetric bilinear form. Proof. The vector space Q of matrices Diagonal[A1, ..., A1; ...; Aj, ..., Aj; ...; As, ..., As] 2 2 considered in Theorem 2.1 has dimension n1 + ... + ns. 2 We consider a new basis for Q whose n1 first vectors are 1 Fij = Diagonal[Eij, ..., Eij;0, ..., 0; ...;0, ..., 0] 2 where i, j 1, ..., n1 . The next n2 vectors in the new basis of Q are ∈{2 } Fij = Diagonal[0, ..., 0; Eij, ..., Eij;0, ..., 0; ...;0, ..., 0] etc.
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