A Note On Monomial Representations Linear Groups by Ben Fairbairn | Papers by Ben

A Note On Monomial Representations Linear Groups BT Fairbairn University of Birmingham, Birmingham, B15 2TT, UK Abstract A matrix is said to be monomial if every row and column has only one non-zero entry. Let G be a group. A representation ρ : G → GLn (C) is said to be a monomial representation of G if there exists a basis with respect to which ρ(g) is a monomial matrix for every g ∈ G. We use elementary methods to classify the irreducible monomial representations of the groups L2 (q), L3 (q) and their natural decorations. 1 Introduction A matrix is said to be monomial if every row and column has only one nonzero entry. Let G be a group. A representation ρ : G → GLn (C) is said to be a monomial representation of G if there exists a basis with respect to which ρ(g) is a monomial matrix for every g ∈ G. Note that since the elements of a finite group have finite order, the non-zero entries these matrices must be roots of unity. Such a representation is therefore writable over the cyclotomic field k Q(ζk ) where ζk satisfies ζk = 1 and ζk = 1. Note that since ζ2 =-1, Q(ζ2 ) = Q. Let p be a prime and q = pr for some positive integer r. In this article we classify the irreducible monomial representations of the groups L2 (q) and their most natural decorations. More specifically, we prove the following. Theorem 1 The only monomial irreducible representations of L2 (q) are • Any linear representations. • Any representation of L2 (2) all of which are writable over Q. • Any representation of L2 (3), the non-trivial linear representations being writable over Q(ζ3 ) and the 3 dimensional representation being writable over Q. • The 7 dimensional representation of L2 (7), writable over Q. • Any irreducible q + 1 dimensional representation of L2 (q), writable over Q(ζd ) where d is some divisor of (q −1)/2 depending on the representation. Monomial representations are computationally useful as they allow group elements to be represented as the product of a permutation and a diagonal matrix making storage, transmission and calculation with group elements much easier. This work has also been motivated by the author’s contribution to the ‘symmetric generation’ programme of Curtis. The existence of an n dimensional monomial representation of a group G enables us to define the action of G on a free group, p n , generated by n elements of order p. This in turn enables us 1 to form the ‘monomial progenitor’ p n : m G. Homomorphic images of progenitors provide enlightening constructions of many groups. For instance, using the ‘sporadic cases’ in theorem 1 the author has been able to produce new presentations of several interesting finite groups. Many of the sporadic simple groups admit extremely elementary constructions via this approach including several of the Mathieu groups, the Held group and the Harada-Norton group. See Curtis [2, p.173, p.277, p.284] for details. This justifies our emphasis on the low dimensional cases since high dimensional linear groups are not involved in the sporadic groups at all owing to their size and the exceptional nature of the representations in the two dimensional case makes them particularly likely to give interesting symmetric generating sets for these groups. Whilst recent work of Hiss and Magaard (personal correspondence) has led to a more general result than the results proven here, their methods are much less elementary. In particular they assume the classification of the finite simple groups, which we do not. Throughout we shall use the standard Atlas notation for groups, [1]. When necessary we shall denote elements of the group L2 (q) by matrices of S := SL2 (q) that are mapped onto the elements of L2 (q) whenZ(S) is factored out. I am is grateful to Rob Curtis, Chris Parker and Kay Magaard for useful discussions and comments on earlier forms of this paper. In Section 2 we give the preliminary results needed to prove Theorem 1. In Section 3 we prove Theorem 1 and in Section 4 we give analogues of Theorem 1 and indicate how they can be proven. In Section 5 we discuss how these results may be adapted to higher dimensions. 2 Preliminaries Monomial representations of a group can be obtained by inducing them from non-trivial linear representations of subgroups by the following theorem. Theorem 2 Let H ≤ G have index n and χ : H → GL1 (C) be a representation of H. Then χ ↑G is a monomial representation of G of dimension n. H See Isaacs [9, p.67] for details. This theorem tells us that to find monomial representations of L2 (q) we need non-perfect subgroups of L2 (q). We shall thus need Dickson’s theorem classifying the maximal subgroups of the groups L2 (q). (Dickson classified all subgroups of L2 (q), but we shall only need the maximal subgroups.) Theorem 3 (Dickson) The maximal subgroups of L2 (q) are • The group A4 when q ≡ ±3 (mod 8), with 5 ≤ q prime; • The group S4 when q ≡ ±1 (mod 8), with either q prime, or q = p2 and 5 ≤ p ≡ ±3 (mod 8); • The groups A5 when q ≡ ±1 (mod 10), with either q prime, or q = p2 and p ≡ ±3 (mod 10). • The dihedral groups of order q − 1 for q ≥ 13 odd and 2(q − 1) for q even; • The dihedral groups of order q + 1 for q =7, 9 odd and 2(q + 1) for q even; 2 • The subfield subgroups: L2 (q0 ), where q is an odd prime power of q0 , for 2 q odd, a prime power of q0 for q even or; P GL2 (q0 ), where q = q0 , for q odd; • The Frobenius group pr : (q − 1)/2 for q odd and pr : (q − 1) for q even, recalling that in Atlas notation, pr denotes an elementary abelian group of order pr ; Dickson’s original theorem is discussed in Dickson [4], but this is difficult to follow. A more modern and readable account may be found in King [10]. Theorem 2 also tells us that to classify the irreducible monomial representations of L2 (q) we need to know the character tables of L2 (q). Generic character tables for these groups are given in Fulton and Harris [7, Section 5.2]. We shall not reproduce the whole character tables here, simply stating the characters when needed. Early work on the irreducible monomial representations was done by Djokovi´ c and Malzan [5], [6] classifying the monomial representations of the symmetric and alternating groups. In particular the exceptional isomorphisms S3 ∼ L2 (2), = A4 ∼ L2 (3), A5 ∼ L2 (4) ∼ L2 (5) and A6 ∼ L2 (9) ensure that whenever we en= = = = counter these groups, their irreducible monomial representations are classified. 3 Proof of the Main Theorem We first prove two lemmas on the linear characters of certain subgroups of L2 (q). Lemma 4 Let q ≥ 5 be odd. The linear characters of the image in L2 (q) of the subgroup of SL2 (q) of upper triangular matrices are those given in Figure 1. Proof. The above are clearly all characters. These (q − 1)/2 characters are the only linear characters as there are only (q − 1)/2 + 2 conjugacy classes and thus (q − 1)/2 + 2 irreducible characters. There are thus only two more irreducible characters and these cannot be linear as 2 q(q − 1)/2 − (q − 1)/2 for q ≥ 5. Lemma 5 Let q be even. The linear characters of the image in L2 (q) of the subgroup of SL2 (q) of upper triangular matrices are those appearing in Figure 2. Proof. The above are clearly all characters of the group. These q − 1 characters are the only linear characters since there are only q conjugacy classes and thus q irreducible characters. There is thus only one more irreducible character and this is not linear as 2 q(q − 1) − (q − 1). Proof of Theorem 1 Any linear representation of L2 (q) is monomial. By Theorem 2 any non linear monomial representation of L2 (q) will be induced from a linear representation of a proper subgroup. From the character tables of the groups L2 (q) the lowest dimensional representations are q + 1 dimensional. We use Dickson’s theorem, Theorem 3, to show that most maximal subgroups, and therefore most subgroups, have index greater than q + 1. Indeed it will turn out that, for most values of q, only one class of subgroups have index equal to q + 1 and can therefore be used to produce monomial representations. Recall that |L2 (q)| = (q − 1)q(q + 1)/hcf (q − 1, 2). We consider each class of maximal subgroups in turn. 3 • A4 : Here q must be odd and satisfy (q − 1)q ≤ 24 and so q = 3 or 5. As |L2 (3)|=6, A4 < L2 (3). As L2 (5) ∼ L2 (4) and |L2 (4) : A4 | = 5, this = subgroup may be considered to be of index q + 1 and these are shown to exist below. • S4 : Here q must be odd and satisfy (q − 1)q ≤ 48 and so q = 5 or 7. As S4 <L2 (5) we are left with the possibility that L2 (7) has a 7 dimensional irreducible monomial representation obtained by inducing up the ‘sign character’ of S4 . Calculating the induced character shows this to be a genuinely new irreducible monomial representation and explicit matrices for this representation are easily produced. • A5 : There are no non-trivial linear characters that can be induced. All proper subgroups of A5 are contained in one of the other subgroups. • Subfield subgroups: The index of subfield subgroups is greater than q + 1 for all q. • Dihedral groups or order (q − 1) or 2(q − 1): These have index q(q + 1)/2 in both even and odd cases. We have that q(q + 1)/2 ≤ (q + 1) if and only if q = 2 where again all the irreducible monomial representations of L2 (2) are known. • Dihedral groups of order (q + 1) or 2(q + 1): In both even and odd cases these subgroups have index q(q − 1)/2. We have that q(q − 1)/2 ≤ (q + 1) if and only if q = 2 or 3 which again are groups for which all irreducible monomial representations are already known. This only leaves the subgroups isomorphic to pr : (q − 1)/2 for q odd or p : (q − 1) for q even, each of index q + 1. First consider the case when q odd. This splits into the cases q ≡ 1 (mod 4) and q ≡ 3 (mod 4) as the character tables are different. If q ≡ 1 (mod 4) then from the character tables we see the only q + 1 dimensional characters are those listed in Figure 3. Using the standard formula for computing induced characters (see [9, p.64]) we find each of the characters χk ↑G has the above form and is thus irreducible. H Moreover it is easy to see that each of these characters arises in this way since the group F× is cyclic and so any automorphism has the form x → xk . q Now if q ≡ 3 (mod 4) then the character tables tell us that the only q + 1 dimensional characters are those given in Figure 4. There are (q − 2)/2 such characters. Again using the standard formula for computing induced characters we find that each of the characters χk ↑G has H the above form and is thus irreducible. Moreover it is easy to see that each of the above characters arises in this way as the group F× is cyclic and so any q automorphism has the form x → xk . Finally if q is even, the only q + 1 dimensional characters are as follows. r # of classes |CG (g)| representative ρ 1 |L2 (q)| 10 01 q+1 L2 (q), q even (q − 2)/2 q−1 xj 0 0 x−j α(xj ) + α(x−j ) 4 1 0 1 q 1 1 1 q/2 q+1 x ∆y yx 0 All symbols used here are as above. There are (q − 2)/2 such characters. Again using the standard formula for computing induced characters we find that each of the characters χk ↑G has the above form and is thus irreducible. H Moreover it is easy to see that each of these characters arises in this way since the group F× is cyclic and so automorphism will be of the form x → xk . q 4 Decorations of L2 (q) Here we give analogues of theorem 1 for SL2 (q), P GL2 (q) and GL2 (q). Note that again, the generic character tables may be found in Fulton and Harris [7, Section 5.2] and the highest dimension of a representation in each case is still q + 1. 4.1 GL2 (q) Theorem 6 No irreducible representations of the groups GL2 (q) are monomial apart from the following. • Any linear representation; • Any representation of GL2 (2) each of which are writable over Q; • The 3 dimensional representation of GL2 (3), writable over Q(ζ4 ); the 2 dimensional representation of GL2 (3) writable over Q(ζ3 ) induced up from the special linear group SL2 (3) and the linear representations, each of which are writable over Q; • Any irreducible q + 1 dimensional representation of GL2 (q), writable over Q(ζd ) where d is some divisor of (q − 1)2 for q depending on the representation. We first note that if q is even then GL2 (q) = Z(GL2 (q)) × SL2 (q) and so the proof is easy. Unless q = 2, any subgroup of GL2 (q) will either be a subgroup of prime index containing the subgroup SL2 (q) or a subgroup of the form Z(GL2 (q)), M where M ≤ SL2 (q) is maximal. The index of any maximal subgroup will therefore either be equal to the index of a maximal subgroup in SL2 (q) or will be a prime less than q − 1. Since the nonlinear representations of GL2 (q) have dimension q − 1, q or q + 1, only the latter can give us faithful monomial representations. Note that if 2|q then L2 (q) = SL2 (q) = P GL2 (q) so the natural analogues of theorem 1 are immediate in these cases. Therefore throughout the rest of this section we shall assume that q is odd. 4.2 SL2 (q) Theorem 7 No irreducible representations of the groups SL2 (q) are monomial apart from the following. • Any linear representation. • Any representation of SL2 (2) all of which are writable over Q. 5 • The 3 dimensional representation of SL2 (3), writable over Q and the linear representations, each of which are writable over Q(ζ3 ). • The 7 dimensional representation of SL2 (7), writable over Q. • Any irreducible q + 1 dimensional representation of SL2 (q), writable over Q(ζd ) where d is some divisor of q − 1 depending on the representation. Since the center of SL2 (q) is contained in every maximal subgroup of SL2 (q), all maximal subgroups may be lifted from those of L2 (q) and will have therefore have the same index. Note that the 7 dimensional representation of SL2 (7) is not faithful since the center is contained in the derived subgroup of the index 7 subgroups of structure 2˙S4 . Similarly the 3 dimensional representation of SL2 (3) is not faithful. 4.3 P GL2 (q) Theorem 8 No irreducible representations of the groups P GL2 (q) are monomial apart from the following. • Any linear representation; • Any representation of PGL2 (2) all of which are writable over Q; • The 2 dimensional representation of PGL2 (3), writable over Q(ζ3 ); the 3 dimensional representations, writable over Q(ζ4 ) and the linear representations, writable over Q; • The 5 dimensional representation of PGL2 (5), writable over Q; • Any irreducible q+1 dimensional representation of P GL2 (q), writable over Q(ζd ) where d is some divisor of q − 1 depending on the representation. The maximal subgroups may be deduced from theorem 3 as P GL2 (q) ≤ L2 (q 2 ). Note that the isomorphisms PGL2 (3) ∼ S4 and PGL2 (5) ∼ S5 make the = = exceptional cases above special cases of the result of Djokovi´ and Malzan [5]. c 5 Higher Dimensions For the groups L3 (q) we have a result analogous to theorem 1. Theorem 9 The only irreducible monomial representations of L3 (q) are • The trivial representations. • Every irreducible q 2 + q + 1 dimensional representation, writable over Q for q odd. The proof is similar to the proof of theorem 1. The maximal subgroups in this case are again listed in King [10, p.6]. The generic character tables were determined by Simpson and Frame [11]. From these data we see that in this case it is sufficient to show that each maximal subgroup (and therefore every subgroup) either has an index greater than (q + 1)(q 2 + q + 1)/hcf (3, q − 1), where hcf (3, q − 1) denotes the highest common factor of 3 and q − 1, or is 6 is a subgroup of this index whose derived subgroup has index 2 (these are the subgroups isomorphic to p2r : P GL2 (q), leading to the distinction between even and odd since P GL2 (2r ) = L2 (2r )). Again, as in the L2 (q) case, variations of this result exist for the various decorations of L2 (q). Note that the exceptional isomorphism L2 (7) ∼ L3 (2) ensures that L2 (7) = should have a 22 + 2 + 1 = 7 dimensional representation, as seen in theorem 1. Note the distinction between the cases n = 2 and n ≥ 3 stems from the fact that the (q n − 1)/(q − 1) dimensional representations are induced up from a subgroups of structure prn : P GLn−1 (q) which has a derived subgroup of index 2 when n ≥ 3. If n = 2 then PGL1 (q) is cyclic giving a greater index. Note that if n = 3 and q = 2 or 3 then these cases also have index 2, despite P GL2 (q) being soluble. The other exceptional cases at n = 2 stem from the exceptional isomorphisms L2 (2) ∼ S3 and L2 (3) ∼ A4 . = = References [1] JH Conway, RT Curtis, SP Norton, RA Parker and RA Wilson “An ATLAS of finite groups”, Clarendon Press 1985 [2] RT Curtis “Symmetric Generation of Groups”, CUP 2007 [3] RT Curtis and S Whyte “The irreducible monomial representations of the covers of the symmetric and alternating groups” preprint, Birmingham 2007 [4] LE Dickson “Linear Groups with an Exposition of the Galois Field Theory”, Leipzig 1901, reprinted Dover 1958 ˇ [5] DZ Djokovi´ and J Malzan “Monomial irreducible characters of the symc metric and alternating groups.”, J. Algebra, 35: 153-158 1975 ˇ [6] DZ Djokovi´ and J Malzan “Imprimitive, irreducible complex characters of c the alternating groups.”, Canad J. Math., 28(6):1199-1204, 1976 [7] W Fulton and J Harris “Representation Theory, A First Course”, SpringerVerlag 1991 [8] RW Hartley “Determination of the ternary collineation groups whose coefficients lie in the GF(2n )”, Ann. of Math. 27 1925/6, 140-158. [9] IM Isaacs “Character Theory of Finite Groups”, Dover 1994 [10] OH King “The subgroup structure of finite classical groups in terms of geometric configurations”, in ‘Survey in Combinatorics, 2005’ (ed BS Webb) CUP 2006. Also available at http://www.staff.ncl.ac.uk/o.h.king/KingBCC05.pdf [11] WA Simpson and J Sutherland-Frame “The Character Tables for SL(3,q), SU(3,q), PSL(3,q), PSU(3,q)” Can. J. Math., Vol. XXV, 3 1973, 486-494. 7