Primitivity

To classify all maximal finite symplectic matrix groups, it suffices to classify only s.i.m.f. matrix groups as Corollary 2.1.10 shows. In this section we will reduce the number of groups to consider even further.

Definition 2.1.13 LetK be a number field. AK-irreducible subgroupG <GL_m(K) is calledprimitive, if G is not conjugate to a subgroup of the wreath product

HoS_k :=

Diag(h₁, . . . , h_k), P ⊗I^m

k |h_i ∈H, P a k×k permutation matrix for some H <GL^m

k(K) where k is a divisor of m.

Similarly, aQ-irreducible symplectic subgroupG < GL_2n(Q) is calledsymplectic prim-itive, ifGis not conjugate to a subgroup ofHoS_k for some H <Sp²ⁿ

k(Q) where k|n.

Remark 2.1.14 A rationally irreducible symplectic subgroup G < GL_2n(Q) is sym-plectic primitive if and only if 4_K(G) is primitive for all minimal totally complex subfieldsK of End(G).

The concept of primitivity is a key ingredient in the determination of all irreducible finite matrix groups. It has some important consequences for normal subgroups.

Theorem 2.1.15 ([NP95, Lemma (III.1)]) Let G <GL_m(K)be a rationally irre-ducible primitive matrix group and N EG. Then hNi_K ≤K^m×m is a simple algebra or equivalently, the naturalKN-moduleK^1×m splits into a direct sum of k isomorphic KN-modules of dimension ^m_k.

Proof: The groupGacts on N by conjugation. Hence it also acts on the set of central primitive idempotents of hNi_K. Thus G permutes the homogeneous components of the natural KN-module K^1×m. But since G is primitive, there can only be one such

component.

Corollary 2.1.16 ([NP95, (III.1)-(III.3)]) Let G < GL2n(Q) be rationally irre-ducible and symplectic primitive. Further let p be a prime divisor of |G|.

(a) If N EG then N ≤Q^2n×2n is a simple subalgebra.

(b) If O_p(G)6= 1 then there exists some k ≥0 such that p^k(p−1) divides 2n.

(c) All abelian characteristic subgroups of Op(G) are cyclic.

Proof:

(a) Let K <End(G) be any minimal totally complex subfield. Let {f₁, . . . , f_r} and {e₁, . . . , e_s} be the central primitive idempotents of the enveloping algebras N and hGi_K respectively. Let χ_i denote the character corresponding to a simple hGi_Ke_i module and let L = Q(χ₁, . . . , χ_s) ⊆ K be their character field. Then L/Q is Galois and ei = ^χ_|G|ⁱ⁽¹⁾P

g∈Gχi(g⁻¹)g ∈ hGi_L. Since G is irreducible, {e₁, . . . , e_s} is a Galois orbit under Gal(L/Q). For any 1 ≤ j ≤ r there ex-ists some i such that e_if_j 6= 0. Since f_j ∈ N is fixed under Gal(L/Q), we get that e_if_j 6= 0 for all i, j. The enveloping algebra h4_K(G)i_K is isomorphic to hGi_Ke_i for some i. Now {e_if₁, . . . , e_if_r} is a set of central idempotents of hNi_Ke_i' h4_K(N)i_K. But 4_K(G) is primitive and therefore h4_K(N)i_K is a simple algebra by Theorem 2.1.15. This showsr = 1, since noe_if_j vanishes.

(b) By (a), O_p(G) has a rationally irreducible representation of degree d for some divisor d of 2n. But for any p-group, d is of the formp^k(p−1) for some k≥0.

(c) Any characteristic subgroup U of Op(G) is a normal subgroup of G. Thus by (a), the abelian groupU admits a faithful irreducible representation. Therefore

U is cyclic.

Suppose N is a normal subgroup of an irreducible and symplectic primitive group G <GL_2n(Q). Then the natural characterχofN is sufficient to recover the conjugacy class of N. If N has several Q-irreducible faithfull representations, we will use the phrase “GcontainsN with characterχ” to distinguish the conjugacy classes of matrix groups isomorphic to N.

If ˜N <GL_m(Q) denotes anQ-irreducible constituent of N, we will identifyN with ˜N since the precise notation ˜N⊗I²ⁿ

m is not very handy.

The following theorem of Philip Hall classifies all finite p-groups whose abelian char-acteristic subgroups are cyclic. In particular, together with the above result, this classifies all possibible candidates for the Fitting subgroups of symplectic primitive irreducible maximal finite (s.p.i.m.f.) matrix groups.

Theorem 2.1.17 (P. Hall) If P is a finite p-group with no noncyclic abelian char-acteristic subgroups, then P is the central product of subgroups P1 and P2 where

(a) P₁ is an extraspecial 2-group and P₂ is either a cyclic, dihedral, quasidihedral or generalized quaternion 2-group.

(b) p is odd and P₁ is an extraspecial p-group of exponent p and P₂ is cyclic.

Proof: See for example [Hup67, Satz 13.10, p. 357].

We close this section by showing that symplectic imprimitive matrix groups can easily be recognized. Further, the wreath products of symplectic primitive irreducible max-imal finite (s.p.i.m.f.) matrix groups are usually again maxmax-imal finite symplectic. So we can restrict the classification to s.p.i.m.f. matrix groups.

Definition 2.1.18 LetF be a nonempty family of bilinear forms onRⁿ. A latticeLin Rⁿis calledindecomposablew.r.t. F, ifLcannot be written as a direct sumL=L1⊕L2

where b(L₁, L₂) = {0} for all b ∈ F. A vector x ∈ L is called indecomposable in L w.r.t. F if it cannot be written asx=y+z with y, z ∈L\ {0} and b(y, z) = 0 for all b∈ F.

Theorem 2.1.19 LetF be a family of bilinear forms on Rⁿ that contains at least one positive definite formf. Then each latticeLinRⁿ admits a decompositionL=⊕^k_i=1L_i where each L_i is indecomposable w.r.t. F and b(L_i, L_j) = {0} for all b ∈ F and all 1≤i < j≤k. This decomposition is unique up to permutation of the Li.

Proof: We adapt [Kne02, Satz (27.2)] slightly. Let L =⊕^l_i=1L⁰_i be any decomposition such thatb(L⁰_i, L⁰_j) = 0 for allb ∈ F and alli6=j. Ifx∈Lis indecomposable w.r.t. F then x ∈ L⁰_i for some i. Thus two indecomposable elements x and y with b(x, y) 6= 0 for someb ∈ F are in the same componentL⁰_i. Two indecomposable elementsx, y ∈L are said to be equivalent if and only if there exists some indecomposable elements x = x₁, . . . , x_r = y ∈ L and some b₁, . . . , b_r ∈ F such that b_i(x_i, x_i+1) 6= 0 for all 1≤i < r. This defines an equivalence relation on the set of indecomposable elements of L. Since the equivalence classes give rise to a orthogonal decomposition of the Euclidean space (Rⁿ, f) there are at most nsuch classesK₁, . . . , K_k say. Denote by L_i the sublattice ofL generated byK_i. For 1 ≤i < j ≤k we haveb(L_i, L_j) = {0} for all b∈ F by construction. Further, every nonzerox∈Lcan be written as a finite sum of indecomposable elements in L. If x is decomposable, we find some r, s∈ L such that x=r+sandb(r, s) = 0 for allb∈ F. In particular 0< f(r, r), f(s, s)< f(x, x). Hence this decomposition procedure must end. ThereforeL=⊕^k_i=1Li is a decomposition ofL which has the desired properties. Each componentL_i is indecomposable and contained inL⁰_j for some j.

To proof the uniqueness, assume that all L⁰_j are also indecomposable. For 1 ≤ j ≤ l let I_j = {1 ≤ i ≤ k | L_i ⊆ L⁰_j} and set M_j := ⊕i∈I_jL_i ⊆ L⁰_j. We are done if we can show L⁰_j = Mj for all j since then |Ij| = 1. Let x ∈ L⁰_j. Write x = Pl

i=1xi with x_i ∈M_i ⊆L⁰_i for all i. Since ⊕^l_i=1L⁰_i =Lthis implies x_i = 0 for all i6=j. So M_j =L⁰_j

as claimed.

Remark 2.1.20 Let G < GL_m(Q) be finite and L ∈ Z(G) . Then every auto-morphism in Aut(L,F(G)) permutes the components of the unique indecomposable orthogonal decomposition ofL wrt. F(G).

Hence a finite irreducible subgroupG <Sp_2n(Q) is symplectic primitive if and only if eachL∈ Z(G) is indecomposable w.r.t. F(G).

Lemma 2.1.21 ([Ple91, Proposition II.7]) Let H < Sp_2n(Q) be s.p.i.m.f. such that E := End(H) is a minimal totally complex number field. If the 2-modular trivial Brauer character is no constituent of the natural 2-modular character of H, then the wreath product HoS_k <Sp_2nk(Q) is s.i.m.f. for all k ≥1.

Proof: Since −I_n ∈ H we have End(HoS_k) = {I_k⊗c| c∈ E} 'E and F(H oS_k) = {I_k⊗F |F ∈ F(H)}.

Let L = L₁ ⊕ · · · ⊕ L_k for some L_i ∈ Z(G). View the L_i as a ˜H-module where H˜ is the direct product of k copies of H. By our assumption L_i and L_j have no common p-modular constituent for i 6=j as ˜H-modules. By [Ple78, Theorem I.1] we get Z( ˜H) = {⊕^k_i=1L_i | L_i ∈ Z(H)}. Hence Z(HoS_k) = {⊕^k_i=1L | L ∈ Z(H)}. The result now follows since E'End(HoS_k) is minimal totally complex and H oS_k =

Aut(L,F(HoS_k)) for all L∈ Z(HoS_k).

The assumption on the 2-modular constituents is necessary. The group H :=

h(_{−1 0}^{0 1})i < Sp₂(Q) is s.p.i.m.f. but HoS₂ ≤ Sp₄(Q) is not maximal finite (see Theo-rem 4.3.1). In fact, this is the only example that we will encounter.