Globally Irreducible Representations of Finite Groups and Integral Lattices

Pham Huu Tiep^{∗}

Institute for Experimental Mathematics University of Essen Ellernstr. 29, 45326 Essen, Germany

Geometriae Dedicata 64 (1997), 85 − 123

1. Introduction

Euclidean integral lattices are among the most interesting objects of mathematics. Although since the 19th century these lattices have been receiving intensive investigation of specialists on algebra, combinatorics and number theory, they are still far from being well-studied, even narrow classes as those of all even unimodular lattices. Integral lattices are closely related to

– ordinary, integral and modular representations of finite groups;

– maximal finite subgroups of GL_{n}(ℤ);

– subgroup structure of finite simple groups;

– algebraic number fields and division algebras over them.

To our mind, one of the most significant of the recent contributions to the theory of Euclidean integral lattices is the Elkies–Shioda construction of several new series of lattices (giving in dimensions n, 64 ≤ n ≤ 10^{3}, sphere packings denser than the previously known ones) using Mordell-Weil lattices of elliptic curves over a function field [11], [36]. A powerful group-theoretic approach to these lattices has been suggested by B. H. Gross [18]. This approach leads in turn to the problem of classifying globally irreducible representations (abbreviated as GIR’s) of finite groups, which itself is of great interest for ordinary and modular representation theory of finite groups.

Actually, the phenomenon of global irreducibility was first distinguished by J. G. Thompson in the course of constructing the sporadic finite simple group F_{3} = T h [Tho 1, 2]. An essential ingredient of his construction (see also [Smi 1, 2]) was an even unimodular lattice Λ_{248} of rank 248 with Aut(Λ_{248}) = ℤ_{2} × F_{3}. Thompson observed that (Λ_{248}, F_{3}) is an example of pairs (G, Λ), G a finite group and Λ a torsion

∗ Since Sept. 1996: Department of Mathematics, Ohio State University, 231 W. 18th Avenue, Columbus, OH 43210, U. S. A.

\begin{matrix}c \\ ◯ \end{matrix} 2000 Kluwer Academic Publishers. Printed in the Netherlands.

free ℤG-module of finite rank, which satisfy the following condition:

Λ/pΛ is an irreducible 𝔽_{p}G-module, for every prime p.  (1)

Clearly, if (·, ·) is a G-invariant integral symmetric bilinear form defined on Λ, then after suitable rescaling the lattice (Λ, (·, ·)) is even unimodular [41]. We shall refer to (1) as to Thompson’s condition. Thompson proposed to find all pairs (G, Λ) with this remarkable property. At that moment there were known only three (tensor indecomposable) examples of such pairs (G, Λ); namely, Λ is the root lattice of type E_{8}, the Leech lattice Λ_{24} [Con 1, 2], and the Thompson–Smith lattice Λ_{248}; and G = Aut(Λ). Only in 1989 R. Gow [16] discovered a few new examples of such pairs (G, Λ). In his examples, G is a double cover 2𝔸_{n} (n an odd square) or 2𝕊_{n} (n a double square), and Λ lies in the basic spin module of G.

The notion of global irreducibility introduced by Gross is a generalization of Thompson’s condition. Recall that in case of (1) the commuting algebra

𝕂 = End_{G}(V) = {α ∈ End_{ℚ}(V) | ∀g ∈ G, g · α = α · g} ,

where V = Λ ⊗_{ℤ} ℚ, is just ℚ. Also, the G-module V is absolutely irreducible. Gross suggested considering a more general situation, that where 𝕂 can be an imaginary quadratic field or a quaternion algebra. More precisely, let G be a finite group, V a finite dimensional ℚGmodule with character χ, and 𝕂 = End_{G}(V). Then we have:

LEMMA 1.1. [18]. The following conditions are equivalent. (i) The ℝG-module V ⊗_{ℚ} ℝ is irreducible. (ii) The ℝ-algebra 𝕂 ⊗_{ℚ} ℝ is a division algebra. (iii) \frac{1}{|G|} ∑_{g∈G} \frac{1}{2}(χ(g)^{2} + χ(g^{2})) = 1. (iv) The space of G-invariant symmetric bilinear forms on V ⊗_{ℚ} ℝ

has dimension 1.

This means that if one of the conditions (i) – (iv) formulated in Lemma 1.1 holds, then the ℚ-algebra 𝕂 is isomorphic to ℚ, an imaginary quadratic field ℚ(\sqrt{−d}) with d ∈ ℕ, or a definite quaternion algebra (\frac{a,b}{ℚ}) with a, b ∈ ℚ, a, b < 0 (see [31]). Now we assume that V ⊗_{ℚ} ℝ is irreducible. Let R be a maximal order in 𝕂 and Λ an RG-lattice in V, that is,

a) Λ is a free ℤ-submodule of rank n = dim_{ℚ} V of V;

b) Λ is stable under the multiplication by R and the action of G. If ℘ is a maximal (two-sided) ideal of R, then one can consider the reduced representation

V_{℘} = Λ/℘Λ of G over k_{℘} = R/℘.  (2)

DEFINITION 1.2. [18]. A rational representation V of a finite group G is said to be globally irreducible if V satisfies the conditions of Lemma 1.1 and the following condition

For all maximal ideals ℘ of R the representation V_{℘} of G is irreducible over k_{℘}.

Tautologically, if (G, Λ) satisfies (1), then the ℚG-module Λ ⊗_{ℤ} ℚ is globally irreducible. To illustrate Definition 1.2, we exhibit some further examples.

EXAMPLE 1.3. Let 𝕂 denote the quaternion algebra ramified at {2, ∞} (that is, Hamilton’s quaternion division algebra), R its unique maximal order. Then the unit group R^{∗} of R is isomorphic to SL_{2}(3), and contains the quaternion group Q_{8} as a normal subgroup. Both act globally irreducibly on 𝕂. (Remark that if the ideal ℘ divides 2, then k_{℘} ≃ 𝔽_{4}, and so the reduction modulo ℘ is irreducible. If ℘ does not divide 2, then k_{℘} ≃ M_{2}(𝔽_{r}) for some prime r.) The stable lattice is the root lattice of type D_{4}.

EXAMPLE 1.4. (A. Adler, A. I. Bondal, M. Craig, Elkies, Gross, Tiep). Let p ≡ 3(mod 4) be a prime, G = L_{2}(p). Let ψ be the complex character of a \frac{1}{2}-discrete series representation of G, which has dimension (p − 1)/2. Then G has a GIR V with character ψ + ψ̄. The commuting algebra 𝕂 is isomorphic to the value field of ψ: 𝕂 ≃ ℚ(\sqrt{−p}). The action of a Borel subgroup of G on V guarantees the irreducibility of V_{℘} for ℘ not dividing p. In this case k_{℘} is 𝔽_{r} (respectively, 𝔽_{r^{2}}), where r is the prime lying in ℘ and ℘ ¯℘ = rR (respectively, ℘ = rR). If ℘^{2} = pR, then k_{℘} ≃ 𝔽_{p} and V_{℘} is isomorphic to Sym^{(p−3)/2}U_{2}, where U_{2} is the standard two-dimensional representation of SL_{2}(p) in characteristic p, and so V_{℘} is also irreducible.

A cyclotomic model for the corresponding lattice Λ is due to Adler. Take ξ to be a pth root of unity in ℂ and define Λ = (1−ξ)^{(p+1)/4} ·ℤ[ξ], with the following scalar product

(u, v) = Tr_{ℚ(ξ)/ℚ}(u · v̄),

(v̄ is the complex conjugate to v ∈ ℚ(ξ)).

Another model for the same lattice is found by Elkies: Λ is the Mordell-Weil lattice of the elliptic curve E defined by the equation y^{2} = x^{3} − x over k. Here, k is the function field of the hyperelliptic curve X defined by the equation y^{2} = x^{p} − x over the prime field 𝔽_{p}. (G acts on X by

( \begin{matrix}a & b \\ c & d \end{matrix} ) ∘ (x, y) = (\frac{ax + b}{cx + d}, \frac{y}{(cx + d)^{(p+1)/2}} ) ,

gross.tex; 9/06/2000; 19:36; p.3

and so on Λ.) The lattice Λ has the following invariants: dim Λ = p − 1, det Λ = p^{(p−1)/2}, min Λ = (p + 1)/2, Aut(Λ) = ℤ_{2} × P GL_{2}(p). Observe that Λ is not globally irreducible with respect to Aut(Λ) (look at reduction modulo p), although Λ is globally irreducible with respect to G, the commutator subgroup of Aut(Λ). (The point is that, unlike G, Aut(Λ) acts absolutely irreducibly on V.)

The following consequence of Definition 1.2 has been derived in [18]:

PROPOSITION 1.5. [18]. If V is a globally irreducible representation of G, then every RG-lattice in V has the form αΛ, where α is a fractional ideal of R. The two-sided class group Cl(R) of R acts simply transitively on the set of isomorphism classes of Euclidean ℤG-lattices Λ in V with End_{G}(Λ) = R.

Gross [18] showed how globally irreducible representations allowed us to reconstruct some of the Elkies–Shioda lattices that were constructed using the theory of elliptic curves. He also showed that the class of globally irreducible representations includes

(a) the rational representation of degree p^{n} − 1 of G = Sp_{2n}(p) with 𝕂 = ℚ(\sqrt{−p}) (p ≡ 3 (mod 4), p a prime) related to the Weil representation of G (and considered also in [15]);

(b) the rational representations of degree p^{2n} − 1 of G = Sp_{2n}(p^{2}) with End_{G}(V) = 𝕂, the definite quaternion algebra ramified at p and ∞ (p an odd prime);

(c) the representation V of dimension 2^{n+1} of G = 2_{−}^{1+2n} · Ω_{2n}^{−}(2) on the Barnes–Wall lattices [3] with End_{G}(V) = 𝕂, the division algebra of Hamilton’s quaternions.

Series (a) might be viewed as a generalization of Example 1.4, and series (c) — a generalization of Example 1.3. Very recently, Elkies, Gross and N. Dummigan have realized the lattices corresponding to series (b) as sublattices of Mordell-Weil lattices of certain elliptic curves, cf. [10]. In particular, this realization leads to the estimate min Λ ≥ (p^{n} + 1)/2 for the corresponding (even, unimodular, of rank p^{2n} − 1) lattices. An intriguing link between globally irreducible representations of certain finite groups and models over ℤ of algebraic groups has been discovered in [19]. The automorphism groups of Euclidean lattices associated with these GIR’s are investigated in [Tiep 1 – 3]. For more information about the lattices associated with currently known GIR’s see the table in the end of the paper.

In this article we find all possible globally irreducible representations for certain sporadic finite simple groups and the groups L_{2}(q) and 2B_{2}(q). We shall see that the most interesting integral lattices for these groups are usually realized in globally irreducible modules. Globally irreducible representations coming from

– projective complex representations of other finite groups of Lie type of rank ≤ 2; or

– Weil representations of finite classical groups; or

– basic spin representations of symmetric and alternating groups are classified in forthcoming papers (cf. in particular [Tiep 4 – 7]).

The plan of the article is as follows.

A sufficient condition for an irreducible character ψ ∈ Irr(G) to be globally irreducible is indicated in [18]. In §2 a necessary condition for global irreducibility is given. The inner geometric structure of RGinvariant lattices is studied in [18]. Here we consider the wider class of all ℤG-lattices. This is done in order to show that the automorphism groups of associated lattices are maximal finite subgroups of GL_{n}(ℚ).

In §3 the sporadic groups M_{11}, M_{12}, M_{22}, M_{23}, M_{24}, J_{1}, J_{2}, J_{3}, McL, HS, He, Ru, Co_{3}, Suz and the Tits group ^{2}F_{4}(2)′ are treated. Globally irreducible representations exist for M_{11}, M_{12}, M_{22}, J_{2}, J_{3}, HS, Ru, Suz and ^{2}F_{4}(2)′. For each of these groups all the globally irreducible representations are determined. It is also shown that the groups 2M_{12}·2, 2M_{22} · 2 are ℂ-reducible maximal finite subgroups of GL_{20}(ℚ). The question about such subgroups was raised by H. Zassenhaus (see [32], [30]).

In §4 the groups L_{2}(q) and ^{2}B_{2}(q) are considered. The main result of this section is that globally irreducible representations exist only for L_{2}(q) with q = p^{f}, f ≤ 2, and are of degree q −1 if q = p ≡ −1( mod 4) or q = p^{2} ≡ 1(mod 2) (these GIR’s were first found in [18]), and of degree 2(p − 1), if q = p ≡ 1(mod 4). The only exceptions are the cases q ≤ 9, where L_{2}(q) has other globally irreducible representations. More generally, in §5 we show that a double extension Sp_{2n}(p)·2 of the symplectic group Sp_{2n}(p) with p ≡ 1(mod 4) has a (unique) GIR of dimension 2(p^{n} − 1), with 𝕂 the quaternion algebra ramified at {p, ∞}. As a consequence, we obtain a new series of even unimodular root-free (if p^{n} > 5) lattices of rank 2(p^{n} − 1), where p ≡ 1(mod 4) is a prime.

A summary of currently known globally irreducible representations is given in §6.

2. Global Irreducibility and Characters

We start with the following well known statement (see e.g. [14]).

LEMMA 2.1. Let ψ be an irreducible complex character of G, and let ℘ be a finite place of the number field ℚ(ψ). If the reduction ψ( mod ℘) is an absolutely irreducible Brauer character ρ of G, then a representation W of G with character ψ can be defined over the completion ℚ(ψ)_{ρ}.

From now on we denote the finite field generated by the values of an absolutely irreducible Brauer character ρ over 𝔽_{p} by 𝔽_{p}(ρ).

The following sufficient condition for ψ ∈ Irr(G) to be (a constituent of the character χ of) a GIR is given in [18].

PROPOSITION 2.2. [18]. Let ψ be an irreducible complex character of G.

(i) Assume that ℚ(ψ) = ℚ and ψ(mod p) is an absolutely irreducible Brauer character of G for all rational primes p. Then χ = ψ is the character of a globally irreducible representation V of G.

(ii) Assume that ℚ(ψ) is an imaginary quadratic field, and ψ(mod ℘) is an absolutely irreducible Brauer character of G for all primes ℘ of ℚ(ψ). Then χ = ψ + ψ̄ is the character of a globally irreducible representation V of G.

(iii) Assume ℚ(ψ) = ℚ, that its Frobenius–Schur indicator ind(ψ) is equal to −1, and that, for all rational primes p, either ψ ≡ ρ( mod p) is absolutely irreducible, or ψ ≡ ρ + ρ^{p}(mod p) where ρ is an absolutely irreducible Brauer character with 𝔽_{p}(ρ) = 𝔽_{p^{2}}. Then χ = 2ψ is the character of a globally irreducible representation V of G.

We shall need the following restatement of Lemma 1.1.

LEMMA 2.3. Assume that V is a GIR for a finite group G with character χ. Set 𝕂 = End_{G}(V). Then one of the following statements holds.

(i) 𝕂 = ℚ, χ ∈ Irr(G), ℚ(χ) = ℚ and ind(χ) = 1.

(ii) 𝕂 is an imaginary quadratic field, and χ = ψ + ψ̄ for some ψ ∈ Irr(G) with ℚ(ψ) = 𝕂.

(iii) 𝕂 is a definite quaternion algebra and χ = 2ψ for some ψ ∈ Irr(G) with ind(ψ) = −1.

Proof. We need only consider case (ii) and show that ℚ(ψ) = 𝕂. For, we view V as a 𝕂G-module. This 𝕂-module is torsion free, as 𝕂 is a division algebra. So V is a free 𝕂G-module with character, say, θ. Since 𝕂 is an imaginary quadratic field, the character χ of ℚG-module V is equal to θ + θ̄ and either ℚ(θ) = ℚ, or ℚ(θ) = 𝕂. Now one can set ψ = θ and see that ℚ(ψ) = 𝕂.

Before giving a modular analogue of Lemma 2.3, we recall some notation. Throughout the article the symbol IBr_{p}(G) denotes the set of all absolutely irreducible Brauer characters in characteristic p of G. If χ is an ordinary character of G, then χ(mod p) denotes the Brauer character in characteristic p corresponding to χ, that is, the restriction of χ to p′-classes. Let 𝔽 be a field of an arbitrary characteristic, W a finite dimensional 𝔽G-module. W (and its character φ) is said to be quadratic, resp. symplectic, if W admits a G-invariant non-degenerate quadratic form q, resp. non-degenerate alternating form b (but W admits no non-degenerate quadratic forms, if p = 2!). Sometimes we shall say that φ is of type +, resp. −. If φ is not real-valued, we say that φ is of type ∘. If the characteristic of the field is not 2, then we also write ind(φ) = 1, −1, and 0, respectively.

LEMMA 2.4. Let W be a 𝔽G-module with Brauer character θ, which is a sum of two absolutely irreducible Brauer characters, where char𝔽 = p. Assume that W is of quadratic type, resp. of symplectic type. Then either

(i) θ = φ_{1} + φ̄_{1} for some φ_{1} ∈ IBr_{p}(G), or

(ii) θ = φ_{1} + φ_{2}, φ_{i} = φ̄_{i} ∈ IBr_{p}(G) and φ_{i} is of quadratic, resp. symplectic type, i = 1, 2.

Proof. One can suppose that 𝔽 is a splitting field for G. Then W has a simple submodule U with Brauer character φ_{1} ∈ IBr_{p}(G). In case W is quadratic, let b denote the bilinear form corresponding to q, the quadratic form defined on W. Set U^{⊥} = {v ∈ W |b(v, U) = 0}. Then dim U^{⊥} = dim W − dim U and W/U^{⊥} ≃ U^{∗} as G-modules. As U is simple, U ∩ U^{⊥} = U or 0. First assume that U ⊆ U^{⊥}. Then U = U^{⊥} and W/U ≃ U^{∗}. This means that θ − φ_{1} = φ̄_{1} ∈ IBr_{p}(G). Now assume that U ∩ U^{⊥} = 0. Then W = U ⊕ U^{⊥} as an orthogonal sum, and the restrictions of q (resp. of b) to U and U^{⊥} are non-degenerate quadratic (resp. symplectic) forms.

The following statement is well known:

LEMMA 2.5. Assume that (i) U is a 𝔽_{p}G-module; (ii) End_{G}(U) contains k = M_{2}(𝔽_{p}); (iii) the kG-module U is irreducible.

Then U is a direct sum of two isomorphic irreducible 𝔽_{p}G-modules.

Proof. Consider a standard basis {e_{11}, e_{12}, e_{21}, e_{22}} for k over 𝔽_{p}. Set U_{i} = e_{ii}U. Since G commutes with k, the U_{i} are 𝔽_{p}G-modules. One directly verifies that U = U_{1} ⊕ U_{2}, and U_{1} and U_{2} are isomorphic irreducible 𝔽_{p}G-modules.

LEMMA 2.6. Assume that a ℚG-module V is ℝ-irreducible, and keep the notation of (2). Assume that a rational prime p is ramified in R: ℘^{2} = pR. Then the k_{℘}G-modules Λ/℘Λ and ℘Λ/pΛ are conjugate.

Proof. We denote by α ↦ ᾱ the canonical anti-involution of 𝕂. Then the (reduced) trace and norm of α are equal to tr(α) = α + ᾱ, N(α) = αᾱ = ᾱα.

First remark that ℘ = \overline{℘}. In fact, by [34] the maximal two-sided ideals ℘ of R and rational primes p are in one-to-one correspondence given by:

pℤ = ℤ ∩ ℘, ℘|pR.

But from the equality pR = ℘^{2} it follows that pR = \overline{℘}^{2} for the maximal two-sided ideal \overline{℘} of R. So \overline{℘} = ℘.

Now we claim that

αa ≡ āα(mod pR)  (3)

for all a ∈ R, α ∈ ℘. For,

αa − āα = (αa + āᾱ) − (āα + āᾱ) = (αa + \overline{αa}) − ā(α + ᾱ).

Here αa ∈ ℘, so \overline{αa} ∈ ℘, αa + \overline{αa} = tr(αa) ∈ ℤ ∩ ℘ = pℤ. Similarly, α + ᾱ ∈ pℤ. Therefore, αa − āα ∈ pR as required.

If ℘ is a principal ideal of R then the statement of the lemma follows immediately from (3). But that is not the case in general, so we have to formalize our arguments. Fix some α ∈ ℘\pR. We state that ℘ = αR + pR. In order to show that we set ℘′ = αR + pR. Then Rα ⊆ ℘′, since by (3) we have aα ≡ αā(mod pR) for all a ∈ R. Now

℘′R = (αR + pR)R ⊆ αR^{2} + pR^{2} = ℘′,
R℘′ = R(αR + pR) = (Rα)R + RpR ⊆ ℘′R + pR = ℘′,

which means that ℘′ R. In particular, pR ⊂ ℘′ ℘. Recall that ℘/pR ≃ R/℘ is a simple algebra. Therefore ℘′ = ℘.

Now we choose a basis {v_{1}, …, v_{n}} of the k_{℘}-space Λ/℘Λ:

Λ = ⟨v_{1}, …, v_{n}, ℘Λ⟩_{R},

and write down the action of g ∈ G on Λ:

gv_{j} ≡ ∑_{i=1}^{n} a_{ij}v_{i}(mod ℘Λ), a_{ij} ∈ R, j = 1, …, n.

We observe that {αv_{1}, …, αv_{n}} form a basis for the k_{℘}-space ℘Λ/pΛ. In fact, any vector from ℘Λ is a sum of vectors of the form βv, β ∈ ℘, v ∈ Λ. Writing

v = ∑_{i=1}^{n} x_{i}v_{i} + ∑_{k} γ_{k}u_{k}, x_{i} ∈ R, γ_{k} ∈ ℘, u_{k} ∈ Λ,

we get

βv = ∑_{i=1}^{n} βx_{i}v_{i} + ∑_{k} βγ_{k}u_{k} ≡ ∑_{i=1}^{n} βx_{i}v_{i}(mod pΛ).

Furthermore, one can write β ∈ ℘ in the form αy +pz for some y, z ∈ R and then by (3) we have

βx_{i} = αyx_{i} + pzx_{i} ≡ \overline{yx_{i}}α(mod pR).

This means that βv ≡ ∑_{i=1}^{n} \overline{yx_{i}}αv_{i}(mod pΛ), as required.

As α ∈ 𝕂, we have

g(αv_{j}) = α.gv_{j} ≡ ∑_{i=1}^{n} αa_{ij}v_{i} ≡ ∑_{i=1}^{n} \overline{a_{ij}}αv_{i}(mod pΛ).

We have shown that if g ∈ G acting on Λ/℘Λ has matrix A = (a_{ij}) in the basis \{v_{1}, …, v_{n}\}, then g acting on ℘Λ/pΛ has matrix Ā = (\overline{a_{ij}}) in the basis {αv_{1}, …, αv_{n}}. In other words, the k_{℘}-modules Λ/℘Λ and ℘Λ/pΛ are conjugate. We remark that the anti-involution α ↦ ᾱ gives an involutory automorphism of k_{℘}, which is trivial when k_{℘} = 𝔽_{p} and the Frobenius automorphism x ↦ x^{p} when k_{℘} = 𝔽_{p^{2}}.

Now we are in a position to give a converse of Proposition 2.2.

PROPOSITION 2.7. Assume that V is a GIR for a finite group G with character χ. Assume that ψ is an irreducible constituent of χ. Then for every rational prime p there is an absolutely irreducible Brauer character ρ such that ψ ≡ e(ρ_{1} +…+ρ_{s})( mod p), where ρ = ρ_{1}, …, ρ_{s} are all distinct conjugates (over \overline{𝔽_{p}}) of ρ. Moreover, either e = 1, or e = 2, 𝕂 is a quaternion algebra and p is ramified in R.

Proof. Suppose that p is a rational prime. Choose a maximal order R in 𝕂 = End_{G}(V), a RG-lattice Λ in V, a maximal two-sided ideal ℘ of R that contains p and consider the reduced representation V_{℘} over k_{℘} (see (2)). We know that k_{℘} is a simple 𝔽_{p}-algebra, isomorphic to 𝔽_{p}, 𝔽_{p^{2}}, or M_{2}(𝔽_{p}). Set ℓ = 𝔽_{p}, 𝔽_{p^{2}}, 𝔽_{p}, correspondingly. By Lemma 2.5, V_{℘} is a sum of 1, 1, or 2 isomorphic irreducible ℓG-modules. Let U denote one of them. Consider a splitting field 𝔼 for G which is a finite normal extension of ℓ. Then U ⊗𝔼 ≃ U_{1}⊕…⊕U_{s}, where U_{1}, …, U_{s} are distinct Gal(𝔼/ℓ)-conjugate absolutely irreducible 𝔼G-modules. Moreover, s is equal to (ℓ(ρ_{1}) : ℓ), where ρ_{i} is the Brauer character afforded by U_{i}, i = 1, …, s (see [23]).

First suppose that 𝕂 is a field. Then we view V as a 𝕂G-module with character ψ. In this case we have ψ ≡ ρ_{1} + … + ρ_{s}(mod p) (and e = 1).

Now we consider the quaternionic case. Here we view V as a ℚGmodule with character 2ψ. If p is unramified, then k_{℘} = M_{2}(𝔽_{p}) and Λ/pΛ is a sum of two isomorphic irreducible 𝔽_{p}G-modules. So as above we have 2ψ ≡ 2(ρ_{1} + … + ρ_{s})(mod p), ψ ≡ ρ_{1} + … + ρ_{s}(mod p). Suppose now that p is ramified. Then k_{℘} = 𝔽_{p^{2}}. If θ is the Brauer character of the 𝔽_{p^{2}}G-module Λ/℘Λ, then the 𝔽_{p}G-module Λ/℘Λ affords the character θ +θ^{p}. By Lemma 2.6, ℘Λ/pΛ has Brauer character θ^{p} (over 𝔽_{p^{2}}), and Brauer character θ + θ^{p} (over 𝔽_{p}). We have shown that there exists ζ ∈ IBr_{p}(G) with 𝔽_{p^{2}}(ζ) = 𝔽_{p^{2k}} such that θ = ζ + ζ^{p^{2}} + … + ζ^{p^{2(k−1)}}. Then we have

2ψ ≡ 2(θ + θ^{p}) = 2(ζ + ζ^{p} + … + ζ^{p^{2k−1}})(mod p).

This means that ψ ≡ ζ + ζ^{p} + … + ζ^{p^{2k−1}}(mod p). Furthermore, e = 1 if 𝔽_{p}(θ) = 𝔽_{p^{2}}, and e = 2 if 𝔽_{p}(θ) = 𝔽_{p}.

The case e = 2 described in Proposition 2.7 can actually occur. For example, one considers the division algebra ℍ of Hamilton’s quaternions as a 4-dimensional rational module for the quaternion group Q_{8} with character 2ψ, where ψ is the unique non-linear irreducible complex character of Q_{8}. Then this module is globally irreducible, and ψ ≡ 2(mod 2). We shall call such primes p with e = 2 exceptional.

We shall need the following slight modification of Proposition 2.7.

PROPOSITION 2.8. Assume that V is a GIR for a finite group G with character χ. Assume that H is a normal subgroup of G and ψ is an irreducible constituent of χ|_{H}. Then for every rational prime p there is an absolutely irreducible Brauer character ρ of H such that ψ ≡ ρ_{1} + … + ρ_{s}(mod p), where ρ_{1}, …, ρ_{s} are some conjugates (over \overline{𝔽_{p}} and under G) of ρ.

Proof. By Proposition 2.7, there exists an irreducible Brauer character θ ∈ IBr_{p}(G) such that χ mod p is a sum of some conjugates of θ. By Clifford’s theorem, each of these conjugates restricted to H is a sum of some G-conjugate irreducible Brauer H-characters. Therefore, ψ mod p is a sum of some conjugate (over \overline{𝔽_{p}} and under G) irreducible Brauer H-characters.

For the rest of this section we assume that V is a GIR for a finite group G. Some information about the inner geometry of RG-lattices is given in the following assertion.

PROPOSITION 2.9. [18]. Let Λ be an RG-lattice in V.

(i) There is a G-invariant, positive definite Hermitian form u ∘ v on V such that the dual lattice

Λ^{⊥} = {v ∈ V |v ∘ L ⊆ R}

coincides with (∏_{℘∈S}) · Λ, where S is a subset of the ramified primes in R. The subset S is determined uniquely by the representation V, and the form ∘ is determined uniquely by the RG-lattice Λ.

(ii) Let (u, v) = tr(u ∘ v) be the canonical positive definite bilinear form on Λ. Then (·, ·) is integral and even on Λ. The dual lattice Λ^{∗} is equal to D^{−1}Λ^{⊥}, where D^{−1} = {α ∈ K|tr(αR) ⊆ ℤ} is the inverse different of R. In particular, when R is tamely ramified (that is, D is squarefree), we have an isomorphism of finite RG-modules

Λ^{∗}/Λ ≃ ⨁_{℘∉S} V_{℘}.

Now we consider an arbitrary, not necessarily R-invariant, ℤG-lattice Λ in V. We suppose that Λ is endowed with some G-invariant positive definite bilinear integral form (·, ·) (which is unique up to scalar, by Lemma 1.1). Let ψ be an irreducible constituent of χ, the G-character afforded by V. We denote by ψ_{p} the reduction of ψ modulo prime p. Set

S = {p|νψ_{p} is of quadratic type},

where either ν = 1, or e = 2 and ν = 1/2, 3/2 (e is defined in Proposition 2.7). We shall suppose that N = dim V ≥ 5.

Note that the inner geometries of RG- and ℤG-lattices need not be the same. For example, consider again the globally irreducible representation of G = ℚ_{8} on the division algebra ℍ of Hamilton’s quaternions. Then ℍ has a unique maximal order R, and Cl(R) = 1. Furthermore, any RG-lattice is isometric to the root lattice of type D_{4}. But some ℤG-lattice is isometric to the root lattice of type 4A_{1}.

PROPOSITION 2.10. There are a suitable rescaling of Λ and a subset S′ ⊆ S with the following properties:

(i) Λ is an even lattice;

(ii) det Λ = ∏_{p∈S′} p^{n(p)}, where n(p) ≥ (1 − ν/2)N;

(iii) S′ is empty if 𝕂 = ℚ;

(iv) Assume p ∈ S′ and that p is not exceptional. Then either n(p) ≡ 0(mod N) and Aut(Λ) stabilizes some lattice Λ′ with (p, det Λ′) = 1, or n(p) ≡ N/2(mod N) and there exists a homomorphism

ϕ_{p} : Aut(Λ) → O_{n}^{ϵ_{1}}(p) × O_{n}^{ϵ_{2}}(p).

Here, n = N/2 and ϵ_{i} = ±1. Furthermore, the kernel K_{p} of ϕ_{p} is an elementary abelian p-group if p > 2, a 2-group of solvability class ≤ 3 and exponent ≤ 4 if p = 2.

Proof. 1) Suppose that det Λ = (Λ^{∗} : Λ) = M. Then Λ^{∗} ⊇ Λ ⊇ MΛ^{∗}. Choose maximal natural k and minimal natural ℓ with property kΛ^{∗} ⊇ Λ ⊇ ℓΛ^{∗}. Then k divides ℓ, and one can rescale Λ by replacing (u, v) to (u, v)/k. So without loss of generality one can suppose that k = 1. We remark at once that Λ is even. In fact, if Λ is an odd lattice, then

Λ_{(2)} = {v ∈ Λ|(v, v) ∈ 2ℤ}

is a sublattice of index 2 in Λ containing 2Λ. This means that 𝔽_{2}Gmodule Λ/2Λ has a trivial composition factor, which contradicts Proposition 2.7 (and the assumption that N ≥ 5).

Now we suppose that a rational prime p divides M = det Λ. Then p divides ℓ, since M divides (Λ^{∗} : ℓΛ^{∗}) = ℓ^{N}. On 𝔽_{p}G-module W = Λ/pΛ we define G-invariant symmetric bilinear form (x̄, ȳ)_{p} = (x|y)(mod p) for x̄ = x+pΛ, ȳ = y+pΛ. As p divides M, this form is degenerate. This means that its kernel (Λ ∩ pΛ^{∗})/pΛ is nontrivial: Λ ⊇ Λ ∩ pΛ^{∗} ⊃ pΛ. Remark that Λ ≠ Λ ∩ pΛ^{∗}, since k = 1 is chosen to be maximal. So (Λ ∩ pΛ^{∗})/pΛ is a nontrivial proper submodule in W. In particular, 𝕂 ≠ ℚ.

One can see that (·, ·)_{p} considered on W_{1} = Λ/(Λ ∩ pΛ^{∗}) is a nondegenerate symmetric bilinear G-invariant form. When p is odd, this immediately implies that W_{1} is of quadratic type. Suppose that p = 2. Then (·, ·)_{2} is a non-degenerate alternating form. We have shown that Λ is an even lattice. Now we show that Λ′ = Λ ∩ 2Λ^{∗} is doubly even, i. e. (v, v) ∈ 4ℤ for all v ∈ Λ′. For, we observe that

(u + v, u + v) = (u, u) + (v, v) + 2(u, v) ≡ 0(mod 4)

if u, v ∈ Λ′ and (u, u) ≡ (v, v) ≡ 2( mod 4). This means that Λ_{4} = {u ∈ Λ′|(u, u) ∈ 4ℤ} is a sublattice of index ≤ 2 in Λ′. Therefore, if Λ′ is not doubly even, then ψ_{2} has the trivial character as a constituent, contrary to Proposition 2.7 (and the assumption N ≥ 5). So Λ′ is doubly even. In particular, the map q : ū ∈ W_{1} ↦ \frac{1}{2}(u, u)(mod 2) is correctly defined on W_{1}, and it gives a quadratic form which alternating form coincides with (·, ·)_{2}. We have proved that W_{1} is of quadratic type for any prime p. Therefore ϑ is real-valued; furthermore, 𝔽_{p}(ϑ) = 𝔽_{p}.

First we suppose that the integer e defined in Proposition 2.7 is equal to 1. Then 𝔽_{p}G-module W has at most two composition factors, and ϑ is equal to ψ_{p}. Therefore p ∈ S and ν = 1. Now suppose that p is exceptional. In this case, 𝔽_{p}G-module W has precisely four composition factors, each of which has Brauer character \frac{1}{2}ψ_{p}. So ν can take one of the values \frac{1}{2}, 1, \frac{3}{2}, and p ∈ S. Finally, we remark that M = det Λ is divisible by

(Λ^{∗} : (Λ + pΛ^{∗})) = \frac{(Λ^{∗} : pΛ^{∗})}{((Λ + pΛ^{∗}) : pΛ^{∗})} = \frac{p^{N}}{(Λ : Λ ∩ pΛ^{∗})} = p^{N−νN/2}.

This means that n(p) ≥ (1 − ν/2)N.

2) Now we assume that e = 1 and deduce assertion (iv) by induction on n(p). Here W_{1} and W_{2} = (Λ∩pΛ^{∗})/pΛ are simple 𝔽_{p}G-modules. We consider the following G-invariant symmetric form

(ū, v̄)′_{p} = \frac{1}{p}(u, v)(mod p)

defined on W_{2}. In general this form need not be non-degenerate.

First suppose that (·, ·)′_{p} is non-degenerate. When p is odd, W_{1} and W_{2} are both of quadratic type. Suppose that p = 2. In 1) we showed that q is a non-degenerate quadratic form on W_{1}, and Λ ∩ 2Λ^{∗} is a doubly even lattice. In particular, (·, ·)′_{2} is a non-degenerate alternating form on W_{2}, and one can define a map

q′ : ū ∈ W_{2} ↦ \frac{1}{4}(u, u)(mod 2).

This definition is correct, because for u ∈ Λ ∩ 2Λ^{∗}, v ∈ Λ we have (u + 2v, u + 2v) − (u, u) = 4(u, v) + 4(v, v) ∈ 8ℤ. Furthermore, the alternating form associated with q′ is just (·, ·)′_{2}. So W_{2} is quadratic, too.

Now we consider the natural homomorphism

τ_{p} : Aut(Λ) → GL(W).

Remark that τ_{p} is faithful if p > 2, and Kerτ_{p} is an elementary abelian 2-group if p = 2. For, let M denote the set of minimal vectors of Λ. Because of the irreducibility of V, one has ⟨M⟩_{ℚ} = V. Take v ∈ M, g ∈ Kerτ_{p}. Then g(v) − v = pu for some u ∈ Λ. Suppose that u ≠ 0. Then p^{2}(v, v) ≤ p^{2}(u, u) = (g(v) − v, g(v) − v). But

(g(v)−v, g(v)−v)+(g(v)+v, g(v)+v) = 2(g(v), g(v))+2(v, v) = 4(v, v),

hence (g(v) + v, g(v) + v) ≤ (4 − p^{2})(v, v). This shows that when p > 2 we must have g(v) = v for all v ∈ M, therefore Kerτ_{p} = 1. Suppose that p = 2. Then under the assumption u ≠ 0 we have g(v) + v = 0. This implies that g(v) = ±v for all v ∈ M. In particular, g^{2} = 1 for all g ∈ Kerτ_{2}, and so Kerτ_{2} is an elementary abelian 2-group.

In fact, τ_{p} maps Aut(Λ) into the group

\{( \begin{matrix}A & C \\ 0 & B \end{matrix} ) | A ∈ O(q), B ∈ O(q′), C ∈ M_{N/2}(𝔽_{p})\} ,

where q(ū) = (ū, ū)_{p} and q′(ū) = (ū, ū)′_{p} if p > 2. Let ϕ_{p} denote the induced homomorphism Aut(Λ) → O(q) × O(q′). Then, clearly, Kerϕ_{p}/Kerτ_{p} is an elementary abelian p-group.

We have established the existence of a homomorphism ϕ_{p} with the required properties. In order to show that n(p) = N/2 we consider the p-adic lattices Γ = Λ ⊗_{ℤ} ℤ_{p} and Γ^{∗} = Λ^{∗} ⊗_{ℤ} ℤ_{p} (here ℤ_{p} is the ring of p-adic integers). Choose N/2 vectors e_{1}, …, e_{N/2} such that ⟨Γ∩ pΓ^{∗}, e_{1}, …, e_{N/2}⟩ = Γ, and set Γ_{1} = ⟨e_{1}, …, e_{N/2}⟩_{ℤ_{p}}. As (·, ·)_{p} is nondegenerate, Γ can be written in the form Γ_{1} ⊕Γ_{2}, where Γ_{2} = Γ_{1}^{⊥} is the orthogonal supplement to Γ_{1} in Γ. Then we have Γ ∩ pΓ^{∗} = pΓ_{1} ⊕ Γ_{2}.

Because (·, ·)′_{p} is non-degenerate, the determinant of the lattice \frac{1}{p}Γ_{2} is a p-adic unit. From this it follows that det Γ = p^{N/2}u, where u is a p-adic unit. This means that n(p) = N/2.

Next we suppose that (·, ·)′_{p} is degenerate. As W_{2} is simple, this form is trivial. This means that (Λ′, (·, ·)) is an Aut(Λ)-invariant integral lattice, where Λ′ = \frac{1}{p}(Λ ∩ pΛ^{∗}). Clearly, Λ′ ⊃ Λ and (Λ′ : Λ) = |W_{2}| = p^{N/2}. Therefore, det Λ′ = p^{−N} det Λ and n(p) ≥ N. Now one can apply the induction hypothesis to the lattice Λ′.

The following statement is well known in the ordinary case:

LEMMA 2.11. Let G = 2 · A be a central extension of a finite group A, H = H^{+} = G · 2, and H^{−} = (2 · A · 2)^{−} the isoclinic variant of H. Let 𝔽 be an algebraically closed field of characteristic other than 2. Let V be a faithful irreducible 𝔽H-module, with character χ^{+}. Then V can be viewed as a faithful irreducible 𝔽H^{−}-module with character χ^{−}. Moreover, ind(χ^{+}) + ind(χ^{−}) = ∑_{i=1}^{κ} ind(α_{i}), if χ^{+}|_{G} = α_{1} + … + α_{κ} is the sum of κ = 1, 2 irreducible G-characters.

Proof. Let Φ : H → GL(V) denote the matrix representation of H on V. Choose γ ∈ 𝔽 such that γ^{2} = −1. Then H^{−} is generated by the endomorphisms Φ(g), g ∈ G, and γΦ(h), h a fixed element in H \G. Let Φ^{−} denote the representation of H^{−} on V. Clearly, Φ^{−} is irreducible.

1) First we consider the case κ = 1. Then Φ|_{G} is irreducible. If ind(α_{1}) = 0, then ind(χ^{+}) = ind(χ^{−}) = 0, and we are done. Let ind(α_{1}) = ϵ with ϵ = ±1. Then Φ|_{G} stabilizes a non-degenerate bilinear form on V, with matrix X and ^{t}X = ϵX. Clearly, Φ|_{G} also stabilizes the bilinear form with matrix Y = ^{t}Φ(h)XΦ(h), hence Y = λX. But h^{2} ∈ G, therefore X = ^{t}Φ(h)Y Φ(h) = λ^{2}X, i.e., λ = ±1. If λ = 1, then Φ stabilizes the bilinear form with matrix X, which implies ind(χ^{+}) = ind(α_{1}) and ind(χ^{−}) = 0. If λ = −1, then Φ^{−} stabilizes the bilinear form with matrix X, which implies ind(χ^{−}) = ind(α_{1}) and ind(χ^{+}) = 0.

2) Now we consider the case κ = 2. Then by Clifford’s Theorem V = V_{1} ⊕ V_{2}, where V_{i} affords the G-character α_{i}, V_{2} = h(V_{1}), and ind(α_{1}) = ind(α_{2}) = ϵ.

Assume ϵ = ±1. Write Φ(h) = ( \begin{matrix}0 & E \\ N & 0 \end{matrix} ), (E denotes the iden-

tity matrix), Φ(g) = diag(A, B) in a suitable basis of V, g ∈ G. Then Φ|_{G} stabilizes a non-degenerate bilinear form on V, with matrix diag(X, Y) and ^{t}X = ϵX, ^{t}Y = ϵY. Since this form is stable under Φ(h^{−1}gh) = diag(B, NAN^{−1}), we get ^{t}BXB = X. But h^{2} ∈ G, therefore ^{t}NXN = X. Now it is clear that Φ stabilizes the bilinear form with matrix diag(X, X), and Φ^{−} stabilizes the bilinear form with matrix diag(X, −X). Thus ind(χ^{+}) = ind(χ^{−}) = ϵ.

Finally, suppose ϵ = 0. If ind(χ^{+} = ind(χ^{−}) = 0, we are done. Suppose for instance ind(χ^{+}) = ν ≠ 0. Endow V with a non-degenerate H-invariant bilinear form (·, ·). Then V_{i} is totally isotropic relative to (·, ·), hence in a suitable basis we have Φ(g) = diag(A, ^{t}A^{−1}) for

g ∈ G, Φ(h) = ( \begin{matrix}0 & M \\ N & 0 \end{matrix} ), and (·, ·) has the matrix ( \begin{matrix}0 & E \\ νE & 0 \end{matrix} ). Since

Φ(h^{−1}gh) = diag(N^{−1}^{t}A^{−1}N, M^{−1}AM) belongs to Φ(G) for all g ∈ G, we must have M = ν^{t}N^{−1}. Now it is clear that Φ^{−} stabilizes the bilinear form with matrix ( \begin{matrix}0 & E \\ −νE & 0 \end{matrix} ). Thus ind(χ^{+}) = ν, ind(χ^{−}) = −ν.

3. Some Sporadic Groups

The aim of this section is to prove the following result.

THEOREM 3.1. Assume that V is a faithful GIR for a finite group G with character χ, and that G contains a normal subgroup H which is a covering group of one of the sporadic finite simple groups M_{11}, M_{12}, M_{22}, M_{23}, M_{24}, J_{1}, J_{2}, J_{3}, McL, HS, He, Ru, Co_{3}, Suz or the Tits group ^{2}F_{4}(2)′. Then one of the following assertions holds.

(i) H = M_{11}, χ|_{H} = κ(10b + 10c);

(ii) H = 2M_{12}, χ|_{H} = κ(10a + 10b);

(iii) H = 2M_{22}, χ|_{H} = κ(10a + 10b);

(iii)’ H = 3M_{22}, χ|_{H} = κ(45c + 45d + 45c′ + 45d′);

(iv) H = 2J_{2}, χ|_{H} = 2κ(6a + 6b);

(v) H = 3J_{3}, χ|_{H} = κ(18a + 18b + 18c + 18d);

(vi) H = 2HS, χ|_{H} = κ56a, κ ≥ 2;

(vii) H = 2Ru, χ|_{H} = κ(28a + 28b);

(viii) H = 6Suz, χ|_{H} = κ(12a + 12b);

(ix) H = ^{2}F_{4}(2)′, χ|_{H} = κ(26a + 26b). Here κ ∈ ℕ, and the constituents of \frac{1}{κ}χ|_{H} are distinct and algebraically conjugate. Furthermore, in all cases but (iii)’ and (vi), one can take κ = 1 and G = H (or G = H · 2), and then V exists and is uniquely determined for each G. In case (vi) one can take κ = 2 and G = H · 2.

Proof. Under the assumptions of the theorem let ψ be an irreducible constituent of χ|_{H}. Clearly, there is κ ∈ ℕ such that χ|_{H} = κ(ψ_{1} + … + ψ_{h}), where ψ_{1} = ψ, ψ_{2}, …, ψ_{h} are all distinct algebraic conjugates of the character ψ. We apply Proposition 2.8 to find all possibilities for ψ. We shall see that (when κ = 1) one can take G = H or G = H · 2, and then χ|_{H} is uniquely determined. By the Deuring– Noether theorem, V is also uniquely determined. Denote by L the finite simple group H/Z(H). We denote the irreducible complex characters of H by their degrees (and additional Latin letters, when there are several characters of equal degree). The irreducible Brauer characters are denoted similarly, but overlined.

1) L = M_{11}. Here L has trivial Schur multiplier Mult(L) and outer automorphism group Out(L). So H = L. Furthermore, the set Irr(H) of all irreducible complex characters of H contain the following 10 characters 1a, 10a, 10b, 10c, 11a, 16a, 16b, 44a, 45a, and 55a. The decomposition matrices for all primes p are given in [24]. Now we have: 11a(mod 2) = \overline{1a} + \overline{10a}, 45a(mod 2) = \overline{1a} + \overline{44a}, 55a(mod 2) = \overline{1a}+\overline{10a}+\overline{44a}, 16a( mod 3) = \overline{1a}+\overline{5a}+\overline{10a}, 16b( mod 3) = \overline{1a}+\overline{5b}+\overline{10b}, 44a( mod 3) = \overline{5a}+\overline{5b}+\overline{10c}+\overline{24a}, 10a( mod 11) = \overline{1a}+\overline{9a}. The remaining two nontrivial characters 10b, 10c are complex conjugate with the value field ℚ(\sqrt{−2}) over ℚ. Furthermore, their reductions modulo any prime p are absolutely irreducible; and are of type +, ∘, ∘, ∘, where p is 2, 3, 5, 11, respectively. So, χ|_{H} = κ(10b + 10c). Moreover, by Proposition 2.2, one can take κ = 1 and G = H, and then one gets a unique GIR of dimension 20 with 𝕂 = ℚ(\sqrt{−2}).

2) L = M_{12}. Here Mult(L) = ℤ_{2}, Out(L) = ℤ_{2}. So H = L or H = 2L. The double cover 2L has the following irreducible characters 1a, 11a, 11b, 16a, 16b, 45a, 54a, 55a, 55b, 55c, 66a, 99a, 120a, 144a, 176a; 10a, 10b, 12a, 32a, 44a, 44b, 110a, 110b, 120b, 160a, 160b. The first 15 characters belong to Irr(L). The decomposition matrices are given in [24]. Using Proposition 2.8 for p = 2 one can exclude the characters 11a, b, 45a, 54a, 55a, b, c, 66a, 99a, 120a, 176a, 12a, 110a, b, 120b, 160a, b. When p = 3 one excludes 16a, 16b, 144a, and 32a. The prime p = 5 eliminates 44a, b. The remaining two nontrivial characters 10a, 10b are complex conjugate with the value field ℚ(\sqrt{−2}) over ℚ. Furthermore, their reductions modulo any prime p are absolutely irreducible; and are of type +, ∘, ∘, ∘, where p is 2, 3, 5, 11, respectively. So, χ|_{H} = κ(10a + 10b). Moreover, by Proposition 2.2, one can take κ = 1 and G = H = 2M_{12}, and then one gets a unique GIR of dimension 20 with 𝕂 = ℚ(\sqrt{−2}). It is easy to see that the unique GIR of M_{11} extends to a (unique) GIR of 2M_{12}. In turn, the latter GIR can be extended to two GIR’s of 2M_{12} · 2. The corresponding lattices have determinant 2^{10}.

3) L = M_{22}. Here Mult(L) = ℤ_{12}, Out(L) = ℤ_{2}. The full covering group 12L has the following irreducible characters 1a, 21a, 45a, 45b, 55a, 99a, 154a, 210a, 231a, 280a, 280b, 385a (the characters of L); 10a, 10b, 56a, 120a, 126a, 126b, 154b, 154c, 210b, 330a, 440a (the faithful characters of 2L); 56b, 56c, 144a, 144b, 160a, 160b, 176a, 560a (and their conjugates, these characters are faithful for 4L); 21b, 45c, 45d, 99b, 105a, 105b, 210c, 231b, 231c, 330b, 384a (and their conjugates, these characters are faithful for 3L); 66a, 66b, 120b, 126c, 126d, 210d, 210e, 210f, 330c, 384b (and their conjugates, these characters are faithful for 6L); 120c, 120d, 144c, 144d, 336a, 336b, 384c (and their conjugates, these characters are faithful for 12L). The decomposition matrices are given in [24] and [22]. Using Proposition 2.8 for p = 2 one can exclude all irreducible characters of L; all faithful characters of 2L but 10a, b; all faithful characters of 4L; all faithful characters of 3L but 45c, d and 384a; all faithful characters of 6L but 384b; and all faithful characters of 12L but 384c. When p = 3 one eliminates 384a, b, c.

The characters 10a, 10b are complex conjugate with the value field ℚ(\sqrt{−7}) over ℚ. Furthermore, their reductions modulo any prime p are absolutely irreducible; and are of type ∘, ∘, ∘, +, ∘, where p is 2, 3, 5, 7, 11, respectively. So, we get χ|_{H} = κ(10a + 10b). Moreover, by Proposition 2.2, one can take κ = 1 and G = H = 2M_{22}, and then one gets a unique GIR of dimension 20 with 𝕂 = ℚ(\sqrt{−7}). It is easy to see that this GIR of 2M_{22} extends to a GIR of 2M_{22} ·2. The corresponding lattices are of determinant 7^{10}.

The remaining nontrivial characters 45c, 45d and their complex conjugates 45c′, 45d′ are algebraic conjugate with the value field ℚ(\sqrt{−3}, \sqrt{−7}) over ℚ. They have absolutely irreducible reductions modulo any prime p. So, χ|_{H} = κ(45c+45d+45c′+45d′). To get a GIR with κ = 1 one might try to take H = 3M_{22} · 2 and consider the irreducible H-character ϑ = Ind_{3L}^{H}(45c) of degree 90 with ℚ(ϑ) = ℚ(\sqrt{−7}). Then ϑ(mod p) is an absolutely irreducible Brauer character if p ≠ 3, and a sum of two absolutely irreducible Brauer characters with the value field 𝔽_{9} over 𝔽_{3}, which are not conjugate to each other. By Proposition 2.7, this means that H admits no GIR of dimension 180. But in general a certain extension G of 3M_{22} might have a GIR with character χ such that χ|_{3M_{22}} = κ(45c + 45d + 45c′ + 45d′).

4) L = M_{23}. Here Mult(L) = 1, Out(L) = 1. So H = L. H has 17 irreducible characters 1a, 22a, 45a, 45b, 230a, 231a, 231b, 231c, 253a, 770a, 770b, 896a, 896b, 990a, 990b, 1035a, 2024a. The decomposition matrices are given in [24]. Using Proposition 2.8 for p = 2 one can exclude the characters 45a, b, 230a, 231a, b, c, 253a, 770a, b, 990a, b, 1035a, and 2024a. When p = 23 one excludes the remaining characters 22a, 896a, b.

5) L = M_{24}. Here Mult(L) = 1, Out(L) = 1. So H = L. H has 26 irreducible characters. The decomposition matrices are given in [24]. In particular, one can see that none of the nontrivial irreducible complex characters of H satisfies Proposition 2.8 with p = 2.

6) L = J_{1}. Here Mult(L) = 1, Out(L) = 1. So H = L. H has 15 irreducible characters 1a, 56a, 56b, 76a, 76b, 77a, 77b, 77c, 120a, 120b, 120c, 133a, 133b, 133c, 209a. We use the Brauer trees of H described in [21]. There, the numbering of the blocks corresponds to the order of the ordinary characters on [1], and a block B has number n if and only if the ordinary characters of non-zero defect preceding the first character of B lie in n − 1 distinct blocks. Using Proposition 2.8 with p = 19 and the Brauer tree for the principal block, one can exclude the characters 56a, b, 77a, b, 120a, b, c. The Brauer tree for the principal block in characteristic 11 allows us to eliminate 76a, b and 133a, b, c. The Brauer trees for blocks 1 and 3 in prime 5 exclude the two remaining characters 77c and 209a.

7) L = J_{2}. Here Mult(L) = ℤ_{2} and Out(L) = ℤ_{2}. The double cover 2L has 38 irreducible characters. The decomposition matrices are described in [20]. In particular, using these matrices and applying Proposition 2.8 for p = 2, 3, 5, one can exclude all characters of 2L, except two characters 6a, 6b of minimal degree, which are algebraically conjugate. Note that ℚ(6a) = ℚ(\sqrt{5}) and ind(6a) = −1. In particular, the space of H-invariant symmetric bilinear forms on a space with character 6a + 6b has dimension 0. In fact, the Schur index m_{ℚ}(6a) is equal to 2. So H = 2J_{2} and χ|_{H} = 2κ(6a + 6b). Assuming κ = 1, we take G = H · 2 = 2J_{2} · 2 whose character table is printed in [1], and set ϑ = Ind_{H}^{H·2}(6a). Then ϑ ∈ Irr(G), ℚ(ϑ) = ℚ and ind(ϑ) = −1. The reduction ϑ(mod p) is an absolutely irreducible Brauer character of type −, if p = 2, 3, 7. Furthermore, ϑ(mod 5) is a sum of two absolutely irreducible Brauer characters of type − with the value field 𝔽_{25} (over 𝔽_{5}). By Proposition 2.2, there is a (unique) GIR V of G = 2J_{2} · 2 with character 2ϑ, of dimension 24 and with 𝕂, a quaternion algebra ramified at 5 and ∞. Applying Lemma 2.4 and Proposition 2.10, one can see that any ℤG-invariant lattice lying in V is isometric to the Leech lattice. A very elegant explicit quaternionic construction of the Leech lattice has been exhibited in [52].

Using Lemma 2.11, it is not difficult to show that the isoclinic variant G^{−} = (2J_{2}.2)^{−} has also a GIR of dimension 24 with character 2ϑ^{−}, where ϑ^{−} = Ind_{H}^{G^{−}}(6a). But in this case, the reduction ϑ^{−}(mod 5) is of quadratic type, and so a ℤG^{−}-invariant lattice can have determinant 5^{12}. In particular, any RG^{−}-lattice is 5-modular. Such a lattice has been constructed in [28].

8) L = J_{3}. Here Mult(L) = ℤ_{3}, Out(L) = 1. The cover 3L has 55 irreducible characters 1a, 85a, 85b, 323a, 323b, 324a, 646a, 646b, 816a, 1140a, 1215a, 1215b, 1615a, 1920a, 1920b, 1920c, 1938a, 1938b, 2432a, 2754a, 3078a (these characters lie in Irr(L)); 18a, 18b, 153a, 153b, 171a, 171b, 171c, 324b, 1215c, 1215d, 1530a, 1530b, 2736a, 2754b, 2907a, 3060a, 3078b (and their conjugates). We use Brauer trees of 3L described in [21]. The Brauer trees for prime 17 allow us to eliminate characters 1140a, 1215a, b, 1920a, b, c, 2432a, 3078a (block 1), 1215c, d, 2736a, 3078b (block 2). The Brauer trees for prime 19 allow us to exclude characters 324a, 816a, 2754a (block 1), 1530a, b, 2754b, 3060a (block 2). The Brauer tree for prime 5 excludes character 324b (block 6). Using tables of Brauer characters in characteristics 2 and 3 [25] one can exclude all the remaining characters but 18a, b (and their conjugates 18c, d). So we must have H = 3L and χ|_{H} = κ(18a+18b+18c+18d). In case κ = 1 we take G = 3J_{3}·2 and set ϑ = Ind_{H}^{G}(18a). Then ϑ ∈ Irr(G), ℚ(ϑ) = ℚ(\sqrt{−15}). The reduction ϑ( mod p) is an absolutely irreducible Brauer character of type ∘, +, +, ∘, ∘, where p = 2, 3, 5, 17, 19, respectively. This means that G has a (unique) GIR V of dimension 72 with character ϑ + ϑ̄ and 𝕂 = ℚ(\sqrt{−15}). Let Λ be any RG-lattice in V. Since R is ramified at 3 and 5, and ϑ(mod p) is of type + for p = 3, 5, by Proposition 8.6 [18] we have det Λ = 3^{36} · 5^{36}.

9) L = HS. Here Mult(L) = ℤ_{2}, Out(L) = ℤ_{2}. The same arguments as in 8) with use of [25] and [21] allow us to exclude all irreducible characters of 2L but 56a, the unique character of degree 56. So H = 2HS and χ|_{H} = κ56a. Set ψ = 56a. Then ℚ(ψ) = ℚ, ind(ψ) = 1. The reduction ψ(mod p) is absolutely irreducible, if p = 2, 3, 7, or 11. But the reduction ψ(mod 5) is a sum of two absolutely irreducible Brauer characters of type ∘ with the value field 𝔽_{5}. We claim that the Schur index m_{ℚ}(ψ) = 1. In fact, ℚ(ψ) = ℚ implies that m_{ℚ}(ψ) = 1 or 2, by the Brauer–Speiser theorem. Suppose that m_{ℚ}(ψ) = 2. Then there exists an irreducible ℚH-module V with character 2ψ. The commuting algebra 𝕂 = End_{H}(V) is obviously a quaternion algebra over ℚ. The information that we have about ψ tells us that 𝕂 is unramified at ∞ and at any prime p ≠ 5. Therefore, by Hasse’s principle, 𝕂 is globally split, a contradiction.

Thus H has a ℚ-module V with character ψ. But this module cannot be a GIR for any finite group G H. Namely, it is easy to see that such a group G coincides with H, and so G does not satisfy Proposition 2.7 for p = 5. To get a GIR, we have to take κ = 2 and consider G = (2HS.2)^{−}, the isoclinic variant of 2HS · 2 whose character table is printed in [1]. This group has a character ϑ of degree 56 (an extension of 56a), with ℚ(ϑ) = ℚ(\sqrt{−5}). Then all prime reductions of ϑ are absolutely irreducible, and by Proposition 2.2 ϑ gives rise to a (unique) GIR of G of dimension 112 with 𝕂 = ℚ(\sqrt{−5}).

10) L = McL. Here Mult(L) = ℤ_{3}, Out(L) = ℤ_{2}. 3L has 66 irreducible characters 1a, 22a, 231a, 252a, 770a, 770b, 896a, 896b, 1750a, 3520a, 3520b, 4500a, 4752a, 5103a, 5544a, 8019a, 8019b, 8250a, 8250b, 9625a, 9856a, 9856b, 10395a, 10395b (these characters lie in Irr(L)); 126a, 126b, 792a, 1980a, 2376a, 2376b, 2520a, 2520b, 2772a, 4752b, 5103b, 6336a, 6336b, 7875a, 8019c, 8019d, 8064a, 10395c, 10395d, 10395e, 12395a (and their conjugates). Since all the nontrivial characters from IBr_{2}(L) are of even degree, one can exclude characters 231a, 5103a, 8019a, b, 9625a, 10395a, b. The 5-decomposition matrices for L and 3L are computed in [54] and [27]. They allow us to eliminate characters 22a, 252a, 770a, b, 3520a, b, 4752a, 5544a, 9856a, b; 792a, 1980a, 2376a, b, 2520a, 2772a, 6336a, 10395c, d, e. The Brauer trees [21] exclude 4500a (p = 7, block 1); 8250a, b (p = 7, block 2); 4752b, 6336b (p = 7, block 3); 8019c, d, 12375a (p = 7, block 5); 1750a (p = 11, block 1); 5103b, 2520b, 7875a, 8064a (p = 11, block 2). Finally, one can use Proposition 2.8 with p = 3 to exclude the remaining characters 126a, b, 896a, b.

11) L = He. Here Mult(L) = 1, Out(L) = ℤ_{2}. So H = L. This group has 33 irreducible characters 1a, 51a, 51b, 153a, 153b, 680a, 1029a, 1029b, 1275a, 1275b, 1275c, 1920a, 4080a, 4352a, 6272a, 6528a, 7497a, 7497b, 7650a, 7650b, 7650c, 10880a, 11475a, 11475b, 11900a, 13720a, 14400a, 17493a, 20825a, 21504a, 21504b, 22050a, 23224a. The 7-Brauer characters of L are described in [35]. The Brauer tree of the block with defect 1 allows us to exclude 20825a, 22050a.

Before considering the principal block we remark that the 7-Brauer characters of H satisfy the following property

φ(x) ∈ ℤ, ∀x ∈ C, ∀φ ∈ IBr_{p}(H);

if φ, φ′ ∈ IBr_{p}(H), φ(1) = φ′(1), then φ(x) = φ′(x), ∀x ∈ C,

(4)

where C is the set of all involutions. Suppose that ψ is an irreducible constituent of a given GIR of G. Then by Proposition 2.8 we have ψ(mod p) = φ_{1} + … + φ_{t}, where φ_{1}, …, φ_{t} ∈ IBr_{p}(H) are characters of the same degree ψ(1)/t. Furthermore, from (4) it follows that

t divides ψ(x) for all x ∈ C.  (5)

This condition puts strong restrictions on possible degrees of φ_{i}. It allows us to exclude most of the characters lying in the principal block. The remaining characters are excluded by Proposition 2.8 with p = 2, 3 or 5, with use of [21] and [25].

12) L = Co_{3}. Here Mult(L) = 1, Out(L) = 1. So H = L. This group has 42 irreducible characters 1a, 23a, 253a, 253b, 275a, 896a, 896b, 1771a, 2024a, 3520a, 3520b, 4025a, 5544a, 7084a, 8855a, 9625a, 9625b, 20608a, 20608b, 23000a, 26082a, 31625a, 31625b, 31625c, 31878a, 40250a, 57960a, 63250a, 73600a, 80960a, 91125a, 93312a, 129536a, 129536, 177100a, 184437a, 221375a, 226688a, 246400a, 249480a, 253000a, 255024a. The Brauer trees that are described in [21] allow us to exclude characters 275a, 5544a, 91125a, 93312a, 246400a, 249480a (p = 23, block 1); 23000a, 26082a (p = 11, block 1); 4025a, 40250a, 57960a, 73600a (p = 11, block 2); 31625a, 129536b (p = 7, block 1); 63250a, 80960a (p = 7, block 2); 31625b, c, 129536a, 184437a, 253000a (p = 7, block 3); 177100a (p = 5, block 2).

Now we observe that IBr_{3}(H) [25] satisfies condition (4) for C, the union of conjugacy classes 5A, 5B, 7A, 10B (in notation of [1]). Then using (5) one can exclude all the 20 remaining characters.

13) L = Ru. Here Mult(L) = ℤ_{2}, Out(L) = 1. Observe that IBr_{2}(L)

[25] satisfies (4) for C, the set of all elements of order 3, 7, 13 and 15. Then using (5) and the description of Brauer trees given in [21] one can exclude all but 6 characters 28a, b, 7280a, b, 8192a, b. Furthermore, 7280a( mod 3) = \overline{1248a}+\overline{6032a}, 7280b( mod 3) = \overline{1248a}+\overline{6032b}, 8192a(mod 3) = \overline{1248a} + \overline{6944a}, 8192a(mod 3) = \overline{1248a} + \overline{6944b}. So H = 2L and χ|_{H} = κ(28a + 28b). Finally, ℚ(28a) = ℚ(\sqrt{−1}), and 28a(mod p) is an absolutely irreducible Brauer character of type + or ∘, if p = 2 or p is an odd prime. This means that G = H has a GIR V of dimension 56 with 𝕂 = ℚ(\sqrt{−1}). In fact, any Hermitian RG-lattice in V is isometric to the ℤ[\sqrt{−1}]-lattice with the automorphism group 4 · L described in [1]. Considered as a ℤ-lattice, it is unimodular.

14) L = Suz. Here Mult(L) = ℤ_{6}, Out(L) = ℤ_{2}. Observe that IBr_{3}(6L) [25] satisfies condition (4) for C, the set of {3, 13}′-elements. With use of (5) one can then exclude all but 35 characters 15795a, 18954a, 189540a (which are faithful on L); 61236a, b (which are faithful on 2L); 78a, b, 429a, b, 2925a, b, c, d, 5103a, b, c, d, 18954b, c, 160380a, b, 189540b, c (which are faithful on 3L); 12a, b, 4368c, d, 11088a, b, c, d, 61236c, d, e, f. Since IBr_{5}(3L)\IBr_{5}(L) satisfies (4) for C, the set of elements of conjugacy classes 2A, 7A, 11A, one excludes 429a, b, 18954b, c, 160380a, b, 189540b, c. Furthermore, the set of faithful absolutely irreducible 5-Brauer characters of 6L satisfies (4) for the set C of all elements of conjugacy class 11A, therefore we can eliminate 4368c, d, 11088a, b, c, d, 61236c, d, e, f. The known absolutely irreducible 2-Brauer characters of 6L given in [25] are enough to exclude all remaining characters, except 12a, b. So H = 6L and χ|_{H} = κ(12a + 12b). Setting ψ = 12a we have ℚ(ψ) = ℚ(\sqrt{−3}), and the reduction ψ(mod p) is an absolutely irreducible Brauer character of type ∘, −, ∘, ∘, ∘, ∘, where p = 2, 3, 5, 7, 11, 13, respectively. Therefore G = H has a GIR V of dimension 24 with 𝕂 = ℚ(\sqrt{−3}). Any RG-invariant (Hermitian) lattice is isometric the complex Leech lattice (see [1], [13]), and any ℤG-lattice in V is isometric to the Leech lattice.

15) L = ^{2}F_{4}(2)′. Here Mult(L) = 1, Out(L) = ℤ_{2}. Using [20] and Proposition 2.8 with p = 2, one excludes all but characters 26a, b, 78a, 2048a, b. Furthermore, 78a(mod3) = \overline{1a} + \overline{77a}, 2048a, b(mod13) = \overline{674a} + \overline{1374a}. So H = L and χ|_{H} = κ(26a + 26b). Finally, ℚ(26a) = ℚ(\sqrt{−2}), and 26a( mod p) is an absolutely irreducible Brauer character of type +, ∘, ∘, ∘, where p = 2, 3, 5, 13, respectively. Therefore G = H has a GIR V of dimension 52 with 𝕂 = ℚ(\sqrt{−2}). Any RG-invariant Euclidean lattice in V is isometric to the lattice Λ described in [1] with Aut(Λ) = 2F_{4}(2) and of determinant 2^{26}.

Theorem 3.1 is proved.

COROLLARY 3.2. The groups 2M_{12} ·2 and 2M_{22} ·2 can be embedded in GL_{20}(ℚ) as ℂ-reducible maximal finite irreducible subgroups.

Proof. Let G denote any of the groups 2M_{12} · 2, 2M_{22} · 2. Let V be a faithful irreducible ℚG-module of dimension 20. Then one readily checks that V is globally irreducible. It remains to show that Aut(Λ) = G for any ℤG-lattice in V. Set A = Aut(Λ). It is not difficult to show that sol(A) = Z(A) = Z = ℤ_{2}, and soc(A/Z) = L is a non-abelian finite simple group which contains G′/Z.

First suppose that G = 2M_{12}·2. By Proposition 2.10, L is a subgroup of Ω_{10}^{±}(2); furthermore, L contains M_{12}. From this it is easy to see that in fact L = M_{12} and A = G. If G = 2M_{22} · 2, then L is a subgroup of Ω_{10}^{±}(7) that contains M_{22}. Again, one can show that L = M_{22} and so A = G.

In [32] W. Plesken mentioned a question posed by H. Zassenhaus about the minimal n_{0} such that GL_{n_{0}}(ℚ) has ℂ-reducible maximal finite subgroups. Our example shows that n_{0} ≤ 20. In fact, it follows from the results of [30] that n_{0} = 16.

COROLLARY 3.3. Aut(M_{12}) = M_{12} · 2 is a maximal subgroup of O_{10}^{−}(2) = Aut(Ω_{10}^{−}(2)).

Proof. Consider a GIR V of G = 2M_{12}·2 of dimension 20 and a RGlattice Λ in V. By Proposition 2.9, Λ^{∗} = π^{−1}Λ, where π = \sqrt{−2}, and so the quadratic modules Λ/2Λ^{∗} and 2Λ^{∗}/2Λ are isomorphic. Therefore, by Proposition 2.10, ϕ_{2} embeds Aut(M_{22}) = G/Z in O = O_{10}^{−}(2). Suppose that M is a maximal subgroup of O containing G/Z and set Ω = Ω_{10}^{−}(2). From the description of maximal subgroups of Ω given in [1] it follows that M ∩ Ω ⊆ 𝔸_{12}, the alternating group on 12 symbols. If M ∩Ω ≠ 𝔸_{12}, then M ∩Ω = M_{12}, M = G/Z, and we are done. Suppose M ∩ Ω = 𝔸_{12}. Since M/(M ∩ Ω) ≃ ℤ_{2} and G/Z ↛ 𝕊_{12}, we obtain that C_{M}(𝔸_{12}) ≠ 1. The last inequality contradicts the fact that any faithful 10-dimensional representation of 𝔸_{12} over 𝔽_{2} is absolutely irreducible.

4. L_{2}(q) and ^{2}B_{2}(q)

The aim of the section is to prove the following result.

THEOREM 4.1. Assume that V is a faithful GIR for a finite group G with character χ, and that G has a normal subgroup H which is a covering group of L = L_{2}(q) or ^{2}B_{2}(q), where q = p^{f}, p is a prime. Then one of the following assertions holds.

(i) q = p ≡ −1(mod 4), H = L_{2}(q), χ|_{H} = κ(η_{1} + η_{2}), where η_{1}, η_{2} are the only irreducible complex characters of H of degree (q − 1)/2;

(ii) q = p ≡ 1(mod 4), H = SL_{2}(q), χ|_{H} = 2κ(η_{1} + η_{2}), where η_{1}, η_{2} are the only irreducible complex characters of H of degree (q − 1)/2;

(iii) q = p^{2} ≡ 1(mod 2), H = SL_{2}(q), χ|_{H} = 2κη_{i}, where i = 1, 2 and η_{1}, η_{2} are the only irreducible complex characters of H of degree (q − 1)/2;

(iv) q = 3, H = SL_{2}(3), χ|_{H} = 2κϑ_{1}, where ϑ_{1} is the unique ℚvalued irreducible complex character of H of degree 2;

(v) q = 5, H = SL_{2}(5), χ|_{H} = 2κϑ_{1}, where ϑ_{1} is the unique faithful irreducible complex character of H of degree 4;

(vi) q = 9, H = 3𝔸_{6}, χ|_{H} = κ(ρ_{1} + ρ_{2} + ρ_{3} + ρ_{4}), where ρ_{1}, …, ρ_{4} are the only irreducible complex characters of H of degree 3;

(vii) q = 9, H = 6𝔸_{6}, χ|_{H} = κ(μ_{1} + μ_{2} + μ_{3} + μ_{4}), where μ_{1}, …, μ_{4} are the only irreducible complex characters of H of degree 6. Here κ ∈ ℕ. Furthermore, one can take κ = 1, G = H (in the cases (i), (iii) – (v)), and G = H · 2 (in the cases (ii) and (vi)).

Proof. Case A. L = L_{2}(q) and p = 2.

Postponing the case q = 4, we have then H = L, and in the notation of [9] H has q + 1 complex irreducible characters 1, St (of degree q), χ_{i} (of degree q+1, i = 1, …, (q−2)/2), ϑ_{j} (of degree q−1, j = 1, …, q/2). Observe that each nontrivial absolutely irreducible 2-Brauer characters of H has even degree. But q ± 1 is odd, so by Proposition 2.8 one can exclude all χ_{i}’s and ϑ_{j}’s. Furthermore, if r is a prime dividing q+1, then by [4] St = 1+ϑ_{j}( mod r), and so the Steinberg character is eliminated.

Case B. L = L_{2}(q) and p > 2.

Here we suppose that H = L or H = 2L. In the notation of [9] 2L has q + 4 complex irreducible characters 1, St, χ_{i} (of degree q + 1, i = 1, …, (q − 3)/2), ϑ_{j} (of degree q − 1, j = 1, …, (q − 1)/2), ξ_{i} (of degree (q + 1)/2, i = 1, 2), η_{j} (of degree (q − 1)/2, j = 1, 2).

1) Based on [4] we find the decomposition numbers for 2L in characteristic p. (On the p-modular characters of SL_{2}(q) see also [39].) Set

J = {I|I = (i_{1}, …, i_{f}), 0 ≤ i_{j} ≤ p − 1}.

Then 2L has precisely q = p^{f} absolutely irreducible Brauer characters β_{I} parametrized by I ∈ J. For I = (i_{1}, …, i_{f}) the character β_{I} is afforded by the module V_{i_{1}}^{(0)} ⊗ V_{i_{2}}^{(1)} ⊗ … ⊗ V_{i_{f}}^{(f−1)}. Here V_{m} denotes the space of homogeneous polynomials of two indeterminates X, Y of degree m over 𝔽_{q}, and V_{m}^{(i)} is obtained from V_{m} by applying the Galois automorphism λ ↦ λ^{p^{i}}. The p-regular elements of 2L are conjugate to 1, the central involution z, a^{k} and b^{l}, where |a| = q − 1, |b| = q + 1, 1 ≤ k ≤ (q − 3)/2, 1 ≤ l ≤ (q − 1)/2. Fix a faithful character ρ of A = ⟨a⟩ and a faithful character σ of B = ⟨b⟩. For I = (i_{1}, …, i_{f}) ∈ J one defines

Σ(I) = {∑_{j=1}^{f} ε_{j}(p − 1 − i_{j})p^{j−1}|ε_{j} = ±1} ,

Ω(I) = {∑_{j=1}^{f} δ_{j}(i_{j} − 2d_{j})p^{j−1}|δ_{j} = ±1, d_{j} ∈ ℕ, 2d_{j} ≤ i_{j}} .

The following combinatorial fact is mentioned in [4]:

LEMMA 4.2. For r ∈ ℤ the set Δ_{r} = {s|s ∈ Ω(I), r ∈ Σ(I), I ∈ J } coincides with {s ∈ ℤ|s ≡ r(mod 2), |r| + |s| ≤ q − 1}.

Remark at once that for I ∈ J we have:

β_{I}|_{A} = ∑_{s∈Ω(I)} ρ^{s}, β_{I}|_{B} = ∑_{s∈Ω(I)} σ^{s}.

LEMMA 4.3. The decomposition numbers for SL_{2}(q) in characteristic p are as follows.

(i) χ_{k}( mod p) = ∑_{I∈J} d_{kI}β_{I}, where d_{kI} = 1 if {k, q−1−k}∩Σ(I) ≠ ∅, and 0 otherwise.

(ii) ϑ_{k}( mod p) = ∑_{I∈J} d_{kI}β_{I}, where d_{kI} = 1 if {k, q+1−k}∩Σ(I) ≠ ∅, and 0 otherwise.

(iii) ξ_{1,2}(mod p) = ∑_{(q−1)/2∈Σ(I)} β_{I}.

(iv) η_{1,2}(mod p) = ∑_{(q+1)/2∈Σ(I)} β_{I}.

Proof. One has only to repeat the proof of Hauptsatz 9.4 of [4]. We carry it out for case (i). Set

β = ∑_{I:k∈Σ(I)} β_{I}, β′ = ∑_{I:q−1−k∈Σ(I)} β_{I}.

Then β(z) = (−1)^{k}β(1), β′(z) = (−1)^{k}β′(1). Furthermore, by Lemma 4.2 we have

β|_{A} + β′|_{A} = ∑_{s:s∈Ω(I),k∈Σ(I),I∈J} ρ^{s} + ∑_{s:s∈Ω(I),q−1−k∈Σ(I),I∈J} ρ^{s} =

= ∑_{s∈Δ_{k}} ρ^{s} + ∑_{s∈Δ_{q−1−k}} ρ^{s} =

= ρ^{k} + ρ^{−k} + ρ^{k}(1 + ρ^{2} + ρ^{4} + … + ρ^{2(q−2)}) = χ_{k}|_{A}.

Similarly, (β + β′)|_{B} = χ_{k}|_{B}.

2) Now we prove by induction on f ≥ 1 that each of the reductions χ_{k}( mod p) and ϑ_{k}( mod p) contains irreducible Brauer characters of distinct degrees, except for the case of ϑ_{1} and f = 1.

First suppose that f = 1. Then χ_{k}(modp) = β_{k} + β_{q−1−k}, ϑ_{k}(mod p) = β_{k−2} + β_{q−1−k} (if k ≥ 2).

Next suppose that f ≥ 2. Writing k = ∑_{j=1}^{f} a_{j}p^{j−1}, 0 ≤ a_{j} ≤ p − 1, we see that k = ∑_{j=1}^{f}{(p − 1) − (p − 1 − a_{j})}p^{j−1}, and so χ_{k}(mod p), ϑ_{k}( mod p) contain β_{1} = β_{p−1−a_{1},...,p−1−a_{f}} of degree (p−a_{1}) … (p−a_{f}). In particular, if k ≡ 0(mod p), i.e. a_{1} = 0, then β_{2} has degree divisible by p. But the degrees of the characters χ_{k}, ϑ_{k} are coprime to p, so their reductions modulo p contain Brauer characters of different degrees, as required. If a_{1} > 0, then there exists r, 1 ≤ r ≤ f − 1, such that a_{2} = a_{3} = … = a_{r} = p − 1, but 0 ≤ a_{r+1} ≤ p − 2. It is so, because k < (q + 1)/2. Then

k = −{(p − 1) − (a_{1} − 1)} + ∑_{j=2}^{r}{(p − 1) − (p − 1)}p^{j−1}+

+{(p − 1) − (p − 2 − a_{r+1})}p^{r} + ∑_{j=r+2}^{f}{(p − 1) − (p − 1 − a_{j})}p^{j−1}.

This means that χ_{k}(mod p), ϑ_{k}(mod p) contain

β_{2} = β_{a_{1}−1,p−1,...,p−1,p−2−a_{r+1},p−1−a_{r+2},...}.

In particular, if r > 1 then the degree of β_{2} is divisible by p, and we are done. Suppose that r = 1 and β_{1}(1) = β_{2}(1). Then (p − a_{1})(p − a_{2}) = a_{1}(p − 1 − a_{2}), which implies that (a_{1}, a_{2}) = ((p + 1)/2, (p − 1)/2). We have shown that χ_{k}(modp) and ϑ_{k}(modp) contains irreducible Brauer characters of distinct degrees, if k ≢ (p^{2} + 1)/2(mod p^{2}). In particular, the case f = 2 is completed.

Now we suppose that k ≡ (p^{2} + 1)/2(mod p^{2}). Then q − 1 − k ≡ (p^{2} − 3)/2(mod p), and one can apply the arguments explained above to χ_{k}. If f = 3, then a_{3} ≤ (p − 3)/2 and ϑ_{k}(mod p) contains

β_{(p−3)/2,(p−1)/2,p−1−a_{3}}, β_{(p−3)/2,(p−1)/2,a_{3}}

of distinct degrees (p^{2} − 1)(p − a_{3})/4 and (p^{2} − 1)(a_{3} + 1)/4. Assume that f ≥ 4 and write k = (p^{2} + 1)/2 + p^{2}ℓ with ℓ = ∑_{j=1}^{f−2} a_{j+2}p^{j−1}. If ℓ ≢ (p^{2} + 1)/2(mod p^{2}), then one can apply our arguments to ϑ_{ℓ} and ϑ_{k}. If ℓ ≡ (p^{2} + 1)/2(mod p^{2}), then q + 1 − k = (p^{2} + 1)/2 + p^{2}ℓ′, where ℓ′ = p^{f−2} − 1 − ℓ ≡ (p^{2} − 3)/2(mod p^{2}). In this case we can use our arguments to ϑ_{ℓ′} and ϑ_{k}. The induction step is completed.

As a result of this point we can exclude all χ_{k}’s and ϑ_{k}’s, except, possibly, ϑ_{1} when q = p.

3) Observe that St ≡ 1 + η_{1} + η_{2}(mod 2), ξ_{i} ≡ 1 + η_{i}(mod 2). Thus St and ξ_{i} are eliminated by Proposition 2.8 with p = 2. Furthermore, the reductions η_{i}(mod r) are absolutely irreducible for any prime r ≠ p, and also for r = p if in addition f = 1. If f = 2, then

η_{1,2}(mod p) = β_{(p−3)/2,(p−1)/2} + β_{(p−1)/2,(p−3)/2} =
= β_{(p−3)/2,(p−1)/2} + (β_{(p−3)/2,(p−1)/2})^{p}.

If f ≥ 3, then η_{1,2}(mod p) contains

β_{(p−3)/2,(p−1)/2,(p−1)/2,...}, β_{(p−3)/2,(p−3)/2,(p−3)/2,(p−1)/2,(p−1)/2,...}

of degrees (p − 1)(p + 1)^{f−1}2^{−f} and (p − 1)^{3}(p + 1)^{f−3}2^{−f}, that are different from each other.

4) We have shown that if ψ is an irreducible constituent of χ|_{H}, then a) either q = p^{2} and ψ = η_{1}, η_{2}, or

b) q = p and ψ ∈ {ϑ_{1}, η_{1}, η_{2}}.

In case a) we have H = SL_{2}(q), ℚ(ψ) = ℚ, ind(ψ) = −1. By Proposition 2.2, χ|_{H} = 2κψ; in particular, one can take κ = 1 and G = H. As shown in [18], G = SL_{2}(p^{2}) has two GIR’s of dimension p^{2} − 1 with 𝕂, the quaternion algebra ramified at {p, ∞}. These two representations are conjugate by an outer automorphism of G.

Now we consider case b). First suppose that ψ = ϑ_{1}. Apart from ψ, χ|_{H} must contain all algebraic conjugates of ψ, which are ϑ_{k} with k, 1 ≤ k ≤ (p − 1)/2 and (k, p + 1) = 1. But due to 2), these conjugates must coincide with ϑ_{1}. It is easy to see that this is possible if and only if p = 3, 5. If p = 3, then ϑ_{1} is the unique ℚ-valued irreducible character of degree 2 of H = SL_{2}(3). This character gives rise to a GIR of H of dimension 4. Namely, SL_{2}(3) ≃ R^{∗}, the unit group of a (unique) maximal order R of the division algebra ℍ of Hamilton’s quaternions, and the natural representation of SL_{2}(3) on ℍ is globally irreducible. If p = 5, then ϑ_{1} is the unique faithful irreducible character of degree 4 of H = SL_{2}(5). We have ℚ(ϑ_{1}) = ℚ, ind(ϑ_{1}) = −1. Furthermore, the reduction ϑ_{1}(mod p) is an irreducible Brauer character if p = 2, 5, and a sum of two irreducible Brauer characters of degree 2 and with the value field 𝔽_{9} (over 𝔽_{3}) if p = 3. By Proposition 2.2, 2ϑ_{1} is the character of a GIR of H of dimension 8. In fact, ϑ_{1} can be extended to a character ϑ of H · 2 = 2𝔸_{5} · 2. Here ℚ(ϑ) = ℚ(\sqrt{−3}), and all prime reductions of ϑ are irreducible. So H · 2 has a GIR of dimension 8 with character ϑ + ϑ̄. Finally, assume ψ = η_{i}. If p ≡ −1(mod 4), then H = L_{2}(p), χ|_{H} = κ(η_{1} + η_{2}). Furthermore, ℚ(η_{i}) = ℚ(\sqrt{−p}). By Proposition 2.2, H has a (unique) GIR of dimension p − 1, with 𝕂 = ℚ(\sqrt{−p}) (see [18]). The remaining case p ≡ 1(mod 4) will be considered in §5 in a more general context of the symplectic group Sp_{2n}(p).

Case C. L = L_{2}(9).

Recall that Mult(L_{2}(9)) = ℤ_{6}. So in addition to case B we have to consider the cases H = 3L and H = 6L. First suppose that H = 3L. Using the 3-Brauer characters of L one can exclude all faithful irreducible characters of 3L of degree 6. The faithful irreducible characters of degree 9, resp. 15, are excluded by Proposition 2.8 with p = 5, resp. p = 2. There remain only four characters ρ_{i}, 1 ≤ i ≤ 4 of degree 3, which are algebraically conjugate, and generate the value field ℚ(\sqrt{−3}, \sqrt{5}). In the notation of [1] we consider G = 3L · 2_{1}, and set α = Ind_{H}^{G}(ρ_{1}). Then α ∈ Irr(G), ℚ(α) = ℚ(\sqrt{−15}), and all prime reductions of α are absolutely irreducible. By Proposition 2.2 G has a GIR V of dimension 12 with 𝕂 = ℚ(\sqrt{−15}). By [18], one can construct an RG-lattice Λ in V which is invariant under the involutory endomorphism of V induced by the canonical anti-involution of 𝕂. It can be seen that det(Λ) = 3^{6} · 5^{6}, and Aut(Λ) = (2 × 3𝔸_{6}) · 2^{2} is a maximal finite subgroup in GL_{12}(ℚ) (see also [30]). A 3-dimensional reflection representation of 2×3𝔸_{6} over ℚ(\sqrt{−3}, \sqrt{5}) is described in [1]: 2 × 3𝔸_{6} is generated by 45 reflections in 270 = 6 × 45 minimal vectors of Λ.

Similarly, when H = 6L one can exclude the faithful characters of degree 12, and there remain only four characters μ_{i}, 1 ≤ i ≤ 4 of degree 6, which are algebraically conjugate, and generate the value field ℚ(\sqrt{2}, \sqrt{−3}). One might try to consider G = 6L · 2_{1} (in the notation of [1]), and set γ = Ind_{H}^{G}(μ_{1}). Then γ ∈ Irr(G), ℚ(γ) = ℚ(\sqrt{−6}), and the prime reductions γ(mod r) are absolutely irreducible for r ≠ 2. But γ(mod 2) is a sum of two Brauer characters of degree 6 with the value field 𝔽_{2}! (On the other hand, observe that 6L · 2 can be embedded in 6𝔸_{7} · 2, and the latter has a 2-modular globally irreducible lattice of rank 24.)

Case D. L = ^{2}B_{2}(q), q = 2^{2n+1}.

First suppose that q = 8 and H ≠ L. In spite of Mult(L) = ℤ_{2}^{2}, ψ must be a faithful character of 2L, if ψ is an irreducible constituent of χ|_{H}. Now we have Irr(2L)\Irr(L) = {40a, b, c, 56a, b, c, 64b, 104a}, IBr_{2}(L) = \{\overline{1a}, \overline{4a, b, c}, \overline{16a, b, c}, \overline{64a}\}. Furthermore,

40a(mod 2) ≡ 4.\overline{1a} + 2.\overline{4a} + \overline{4b} + 2.\overline{4c} + \overline{16c},
56a(mod 2) ≡ 4.\overline{1a} + 2.\overline{4a} + 2.\overline{4b} + \overline{4c} + \overline{16a} + \overline{16b},
104a(mod 2) ≡ 8.\overline{1a} + 4.(\overline{4a} + \overline{4b} + \overline{4c}) + (\overline{16a} + \overline{16b} + \overline{16c}),

64b(mod 5) ≡ \overline{8a} + \overline{56a}.

We arrive at a contradiction with Proposition 2.8.

Because Mult(L) = 1 if q > 8, now we can suppose that H = L. Set r = 2^{n+1}. In the notation of [5], H has the following irreducible characters: 1, St, Γ_{1,2} (of degree r(q − 1)/2), Ω_{s} (of degree q^{2} + 1, 1 ≤ s ≤ q/2 − 1), Θ_{l} (of degree (q − 1)(q − r + 1), 1 ≤ l ≤ (q + r)/4), Λ_{u} (of degree (q − 1)(q + r + 1), 1 ≤ u ≤ (q − r)/4). If t is a prime dividing q + r + 1, then for the Steinberg character St we have that St ≡ 1 + Γ_{1} + Γ_{2} + Θ_{m}(mod t) for an appropriate m [5], which excludes St. The 2-decomposition numbers for L are found in [5]. In particular, Γ_{1,2}(mod 2) contains the trivial 2-Brauer character with multiplicity 2^{n}. Furthermore, the characters Ω_{s}(mod 2), Θ_{l}(mod 2) and Λ_{u}(mod 2) also contain the trivial Brauer character, since they are of odd degree, meanwhile all the nontrivial 2-Brauer characters are of even degree. This argument completes the analysis of case D.

Theorem 4.1 has been proved.

Now we proceed to describe the automorphism groups of the Euclidean lattices Λ of rank q − 1 associated with GIR’s of S = SL_{2}(q). We restrict ourselves to case (iii) of Theorem 4.1. Case (i) has been treated in [45]; it is shown there that Aut(Λ) ∈ {2 × L_{2}(p), 2 × P GL_{2}(p)}.

Consider case (iii). Recall [18] that if R is a maximal order in the quaternion algebra 𝕂 ramified at {p, ∞}, then there are h_{R} (= 1 or 2) Euclidean ℤG-lattices Λ with End_{S}(Λ) = R. These lattices have invariants: dim Λ = p^{2} − 1, det Λ = 1, min Λ ≥ (p + 1)/2. Now we suppose that V is a GIR for S = SL_{2}(p^{2}) of dimension p^{2} − 1 with character 2ψ, and that Λ is an S-invariant lattice in V. Set G = Aut(Λ).

THEOREM 4.4. Under the above assumptions, one of the following assertions holds.

(i) p = 3, G = W (E_{8}) and Λ is the root lattice of type E_{8}; (ii) p = 5, G = 2Co_{1} and Λ is the Leech lattice; (iii) p ≡ −1(mod 4), p > 3 and Sp_{4}(p) G ⊆ Sp_{4}(p) · 2; (iv) p > 5 and SL_{2}(p^{2}) ∗ ℤ_{ν} G ⊆ ΣL_{2}^{−}(p^{2}) · ℤ_{ν} with ν = 2, 4, or 6.

(Here ∗ denotes the central product with the joint central subgroup ℤ_{2}.)

Proof. 1) First suppose that p = 3 or 5. By Proposition 2.10 Λ is isometric to the root lattice of type E_{8} or the Leech lattice, and we arrive at assertion (i) or (ii). Therefore in what follows we may suppose that p ≥ 7. Set R′ = End_{S}(Λ). Then R is a subring in 𝕂 = End_{S}(V) that contains the identity element. Choose a basis {e_{1}, …, e_{N}} of Λ, N = p^{2} −1. If φ ∈ 𝕂, then there exists m ∈ ℕ such that mφ(e_{i}) ∈ Λ for all i, which implies that mφ ∈ R′. In other words, ℚR′ = 𝕂. Finally, if φ ∈ R′, φ is presented by matrix A in the basis {e_{1}, …, e_{N}}, and f(t) ∈ ℤ[t] is the characteristic polynomial of A, then f(φ) = 0. This means that φ is integral over ℤ. We have shown that R′ is an order in 𝕂. Let R be a maximal order in 𝕂 that contains R′. Since the discriminant of 𝕂 is p and p ≥ 5, the unit group R^{∗} of R is a cyclic group of order 2, 4, or 6 [53]. The same is true for the unit group of R′. In particular, C_{G}(S) is a cyclic group of order ν = 2, 4, or 6.

2) Now we choose some normal subgroup G_{0} in G as follows. If G acts primitively on V ⊗_{ℚ} ℂ, then G_{0} = G. Next suppose that G is either an imprimitive irreducible, or a reducible linear group on V ⊗ ℂ. Then G preserves some nontrivial decomposition V = U_{1} ⊕ … ⊕ U_{s}, s > 1. It is obvious that S fixes each of the U_{i}’s, so s = 2 and U_{i} affords S-character ψ. In this case we set G_{0} = St_{G}(U_{1}, U_{2}). If χ denotes the G-character of V, then χ|_{G_{0}} = ξ + ξ̄ for some ξ ∈ Irr(G_{0}) with ξ_{S} = ψ.

In any case we have s = (G : G_{0}) ≤ 2 and G_{0} ⊇ S. The standard arguments (see e.g. [45]) show that sol(G_{0}) = Z(G_{0}) ⊆ C_{G}(S), and, moreover, L = soc(G_{0}/Z(G_{0})) is a non-abelian finite simple group that contains S̄ = L_{2}(p^{2}). If L = 𝔸_{ℓ}, the alternating group on ℓ symbols, then ℓ ≥ P (S) = p^{2} +1. Here, P (X) (resp. d(X)) denotes the minimum index of proper subgroups (resp. the minimum degree of nontrivial projective complex representations) of a finite group X. Since d(𝔸_{ℓ}) = ℓ−1 for ℓ ≥ 8, we obtain dim V ≥ ℓ−1 = p^{2} > N, a contradiction. If L is a finite group of Lie type defined over a field of characteristic v ≠ p, then the minimum degree of nontrivial projective representations of S, and hence of L, in characteristic v is not smaller than (p^{2} − 1)/2 ≥ 24. By [26], this implies that d(L) ≥ 2^{(p^{2}−1)/4} > N, a contradiction. Suppose that L is one of 26 sporadic simple groups. As |L| is divisible by p^{2}(p^{4} − 1)/2 with p ≥ 7 and d(L) ≤ N, we must have L = Co_{1}. In fact, G_{0} = 2Co_{1}, and \frac{1}{2}χ|_{G_{0}} coincides with the unique irreducible character of degree 24 of 2Co_{1}. In particular, ψ = \frac{1}{2}χ|_{S} can be realized over ℝ, again contradiction. We arrive at the conclusion that L is a finite group of Lie type in characteristic p. Using [26], one can show that in fact L is either P Sp_{4}(p) or L_{2}(p^{2}).

3) Here we suppose that L = P Sp_{4}(p). Then Mult(L) = ℤ_{2}, Out(L) = ℤ_{2}. It is well known that if ξ ∈ Irr(2L)\{1} and ξ(1) ≤ p^{2} − 1, then ξ(1) = (p^{2} ± 1)/2, and ξ is character of an irreducible constituent of the Weil representation. This means that G_{0} 2L = Sp_{4}(p), and χ|_{2L} is a sum of two irreducible characters ξ_{i} of degree (p^{2} − 1)/2, i = 1, 2. The case p ≡ 1(mod 4) is impossible, because then ℚ(ξ_{i}) = ℚ(\sqrt{p}) and the Schur index m_{ℝ}(ξ_{i}) is equal to 2. In the case p ≡ −1(mod 4) we are dealing with the situation studied in [15], [18], [44]. In particular, one has Sp_{4}(p) G ⊆ Sp_{4}(p) · 2.

4) Finally, suppose that L = L_{2}(p^{2}). Then, clearly, SL_{2}(p^{2})∗ℤ_{ν} G ⊆ (SL_{2}(p^{2}) ∗ ℤ_{ν}) · 2^{2}. Observe that the characters ψ = η_{1}, η_{2} of minimal degree (p^{2} − 1)/2 of S are invariant under the field automorphism 2_{F}. The two other automorphisms transpose η_{1} and η_{2}. This means that S · 2^{2} ↛ G. Remark that there are two isoclinic groups of type 2L · 2_{F}. One of them is ΣL_{2}(p^{2}), the group of two-dimensional semilinear transformations over 𝔽_{p^{2}} with determinant 1. Furthermore, this group can be embedded in Sp_{4}(p), and so it has two irreducible characters ψ̃_{1,2} of degree (p^{2}−1)/2 such that ψ̃_{i}|_{S} = ψ and ℚ(ψ̃_{i}) ⊆ ℚ(\sqrt{ϵp}), where i = 1, 2 and ϵ = (−1)^{(p−1)/2}. In fact, ΣL_{2}(p^{2}) contains an element x of order 2p lying in the conjugacy class D_{32} (in the notation of [40]) of Sp_{4}(p). Therefore, ψ̃_{i}(x) = ±\sqrt{ϵp}, and so ℚ(ψ̃_{i}) = ℚ(\sqrt{ϵp}). Considering the action of Gal(ℚ(\sqrt{ϵp})/ℚ), one can show that ψ̃_{i}(y) lies in ℚ if y ∈ S, and in ℚ\sqrt{ϵp} if y ∈ ΣL_{2}(p^{2})\S. From this it follows that for the other group of type 2L·2_{F} we have ℚ(ψ̃_{i}) = ℚ(\sqrt{−ϵp}) (cf. Lemma 2.11). Now we denote the group of type 2L · 2_{F} with ℚ(ψ̃_{i}) = ℚ(\sqrt{p}) by ΣL_{2}^{+}(p^{2}), and the other group by ΣL_{2}^{−}(p^{2}). If ΣL_{2}^{+}(p^{2}) can be embedded in G, then the restriction of χ to this group coincides with ψ̃_{1} + ψ̃_{2}. This implies that the Schur index m_{ℝ}(ψ̃_{i}) is equal to 1. In particular, ψ can be realized over ℝ, a contradiction. Therefore we must have SL_{2}(p^{2}) ∗ ℤ_{ν} G ⊆ ΣL_{2}^{−}(p^{2}) ∗ ℤ_{ν}.

Conversely, the characters ψ̃_{i} of ΣL_{2}^{−}(p^{2}) lead to a GIR of ΣL_{2}^{−}(p^{2}) of dimension p^{2} − 1 with 𝕂 = ℚ(\sqrt{−p}). For, the reductions ψ̃_{i}(mod r) are absolutely irreducible when r ≠ p, as ψ(mod r) is absolutely irreducible. Furthermore, the field automorphism 2_{F} transposes two absolutely irreducible components of ψ(mod p), so ψ̃_{i}(mod p) is also absolutely irreducible.

Theorem 4.4 has been proved.

REMARK 4.5. Similar arguments are applicable to the Sp_{2n}(p^{2})-inva-riant lattices of rank p^{2n} − 1 discovered in [18], for an arbitrary n.

As regards the integer ν defined in Theorem 4.4, one has the following observation:

LEMMA 4.6. Let 𝕂 = (\frac{a,b}{ℚ}) be the quaternion algebra ramified at {p, ∞}, where p > 3 is a prime. Then 𝕂 has a maximal order R with the unit group R^{∗} isomorphic to ℤ_{4} (resp. ℤ_{6}) if and only if p ≡ −1( mod 4) (resp. p ≡ −1(mod 6)).

Proof. We consider the standard basis {1, i, j, k} (see [31]) for 𝕂. Suppose that 𝕂 ∋ λ = r+αi+βj+γk with λ^{2} = −1, but p ≡ 1( mod 4). Then r = 0, and −1 = δ^{2} for some δ ∈ ℚ_{p}^{∗}. Now we have N(δ + αi + βj + γk) = 0, which implies that 𝕂 splits over ℚ_{p}, a contradiction.

Similarly, if 𝕂 ∋ λ = r + αi + βj + γk with λ^{3} = 1 but p ≡ 1(mod 6), then r = −1/2 and −3/4 = δ^{2} for some δ ∈ ℚ_{p}^{∗}. In this case we have N(δ + αi + βj + γk) = 0, which means that 𝕂 splits over ℚ_{p}, again contradiction.

Conversely, if p ≡ −1(mod 4), then one can take 𝕂 = (\frac{−1,−p}{ℚ}). It is easy to show that R = ⟨1, i, (1 + j)/2, (i + k)/2⟩_{ℤ} is an order in 𝕂 with discriminant p. Therefore R is a maximal order in 𝕂, and obviously R^{∗} = ℤ_{4}. Similarly, if p ≡ −1(mod 6), then one can take 𝕂 = (\frac{−3,−p}{ℚ}), R = ⟨j, (1 + i)/2, (j + k)/2, (3 + i + 2k)/6⟩_{ℤ}.

5. One More Globally Irreducible Representation for Symplectic Groups

The aim of this section is to construct one more example of globally irreducible representations for finite symplectic groups.

Throughout this section we fix a prime number p ≡ 1(mod 4) and consider some subgroups G, H of the symplectic group T = Sp_{2n}(p^{2}). In the natural 2n-dimensional 𝔽_{p^{2}}-module W of T consider a symplectic basis {e_{1}, …, e_{n}, f_{1}, …, f_{n}}, that is, a basis in which the symplectic form is given as follows: (e_{i}, e_{j}) = 0, (f_{i}, f_{j}) = 0, (e_{i}, f_{j}) = δ_{i,j}. In this

basis we set U = ⟨e_{1}, …, e_{n}, f_{1}, …, f_{n}⟩_{𝔽_{p}}, J = ( \begin{matrix}εE_{n} & 0 \\ 0 & ε^{−1}E_{n} \end{matrix} ). Here

ε ∈ 𝔽_{p^{2}} is chosen with order 2(p − 1). Now we set H = T ∩ End(U) ≃ Sp_{2n}(p), G = ⟨H, J⟩. Then J^{2} ∈ H and J normalizes H, therefore G = Sp_{2n}(p) · 2. Suppose that g ∈ C_{G}(H)\H. Since the action of H on W is absolutely irreducible, g = λE_{2n} for some λ ∈ 𝔽_{p^{2}}. Now from the inclusion g ∈ G\H it follows that λ^{2} = 1 and λε ∈ 𝔽_{p}, which is impossible because ε ∉ 𝔽_{p}. We have shown that Z(G) = Z(H) = ℤ_{2}. In other words, G = 2L · 2 for L = P Sp_{2n}(p).

Let η_{1,2} denote the two irreducible characters of degree (p^{n} −1)/2 of H. It is well known that η_{i} is faithful, ℚ(η_{i}) = ℚ(\sqrt{p}) and m_{ℝ}(η_{i}) = −1 (see e.g. [18]). Furthermore, the reductions η_{i}(mod r) are absolutely irreducible for any prime r (see [15], [18]). Due to the well-known result of Thompson [43], the equality m_{ℝ}(η_{i}) = −1 implies that η_{i}(mod r) is of type − for any r ≠ 2. It is recently shown in [17] that η_{i}(mod 2) is also of type −.

It is well known that η_{1}+ξ_{1} is the character afforded by the so-called Weil representation of H, where ξ_{1} ∈ Irr(H) and ξ_{1}(1) = (p^{n}+1)/2. The restriction of the Weil representation to the Siegel parabolic subgroup P of H is described in [18]. Setting B = ⟨e_{1}, …, e_{n}⟩_{𝔽_{p}}, then ⟨f_{1}, …, f_{n}⟩_{𝔽_{p}} is canonically identified with the dual space B^{∗}. Any element X ∈ O_{p}(P) is written in the form X = ( \begin{matrix}E_{n} & A \\ 0 & E_{n} \end{matrix} ), where A : B^{∗} → B

is symmetric. Fix a non-trivial character α of the additive group 𝔽_{p}: α(1) = ζ = exp(\frac{2πi}{p}). The Weil representation is given by the action of H on the space of functions f : B^{∗} → ℚ(ζ). In particular, the action of X ∈ O_{p}(P) is as follows:

(X ∘ f)(b) = α((A(b), b)).f(b), ∀b ∈ B^{∗}.

Denote the central involution of H by I. Then we have

η_{1}(X) + ξ_{1}(X) = ∑_{b∈B^{∗}} α((A(b), b)), η_{1}(IX) + ξ_{1}(IX) = 1.

But I acts trivially, resp. faithfully, in a representation with character ξ_{1}, resp. η_{1}. Therefore −η_{1}(X) + ξ_{1}(X) = 1, and so

η_{1}(X) = \frac{1}{2} ∑_{b∈B^{∗}} α((A(b), b)) − \frac{1}{2}.  (6)

Now we choose X such that A(f_{j}) = δ_{1,j}e_{j}, and denote Y = JXJ^{−1}.

Then Y = ( \begin{matrix}E_{n} & ε^{2}A \\ 0 & E_{n} \end{matrix} ), and from (6) it follows that

η_{1}(X) = \frac{1}{2}p^{n−1} ∑_{a∈𝔽_{p}} ζ^{a^{2}} − \frac{1}{2} = \frac{−1 + p^{n−1}\sqrt{p}}{2},

η_{1}(Y) = \frac{1}{2}p^{n−1} ∑_{a∈𝔽_{p}} ζ^{ε^{2}a^{2}} − \frac{1}{2} = \frac{−1 − p^{n−1}\sqrt{p}}{2}.

Our calculations show that J fuses two conjugacy classes of p-elements, with representatives X and Y, in H, and η_{1}(X) ≠ η_{1}(Y) = η_{2}(X). From this it follows that ψ ∈ Irr(G) and ψ_{H} = η_{1} + η_{2}, where ψ = Ind_{H}^{G}(η_{1}). It is easy to see that ℚ(ψ) = ℚ, ind(ψ) = −1. Furthermore, if r is an arbitrary prime other than p, then J fuses two conjugacy classes of r′-elements with representatives X and Y. Therefore the reduction ψ(mod r) is an absolutely irreducible Brauer character, which is of type −.

Now we consider the reduction ψ(mod p). It is shown in [18] that η_{1,2}(mod p) is obtained by restricting the irreducible algebraic representation of Sp_{2n}(\overline{𝔽_{p}}) with highest weight ω_{n−1} + ((p − 3)/2)ω_{n} to H = Sp_{2n}(p). Passing the embedding H ↪ Sp_{2n}(\overline{𝔽_{p}}) through Sp_{2n}(p^{2}), one sees that η_{1,2}(modp) is extended to two absolutely irreducible Brauer characters φ_{1}, φ_{2} of G. Obviously ψ(mod p) = φ_{1} + φ_{2}. We calculate the value of φ_{1} on the element J. If one denotes ϵ = exp(\frac{πi}{p−1}), then φ_{1}(J) = ∑_{u∈I_{n}^{−}} ϵ^{|u|}. Here

I_{n}^{−} = {u = (u_{1}, …, u_{n})|u_{j} ∈ ℤ, |u_{j}| ≤ (p − 1)/2, ∑_{j=1}^{n} u_{j} ≡ 1(mod 2)} ,

and |u| = ∑_{j} u_{j} for u ∈ I_{n}^{−}. Set

I_{n}^{+} = {u = (u_{1}, …, u_{n})|u_{j} ∈ ℤ, |u_{j}| ≤ (p − 1)/2, ∑_{j=1}^{n} u_{j} ≡ 0(mod 2)} ,

S_{n}^{+} = ∑_{u∈I_{n}^{+}} ϵ^{|u|}, S_{n}^{−} = ∑_{u∈I_{n}^{−}} ϵ^{|u|}.

Denote also θ = cot(\frac{π}{2(p−1)}). Then we have

S_{1}^{+} = \frac{θ − θ^{−1}}{2} , S_{1}^{−} = \frac{θ + θ^{−1}}{2} ,

S_{n+1}^{+} = S_{n}^{+}S_{1}^{+} + S_{n}^{−}S_{1}^{−}, S_{n+1}^{−} = S_{n}^{+}S_{1}^{−} + S_{n}^{−}S_{1}^{+}.

From this it follows that

S_{n}^{+} = \frac{θ^{n} + (−θ)^{−n}}{2} , S_{n}^{−} = \frac{θ^{n} − (−θ)^{−n}}{2} .

In particular, φ_{1}(J) = (θ^{n} − (−θ)^{−n})/2 is a positive real number. Moreover, the Galois automorphism ϵ ↦ ϵ^{p} sends φ_{1}(J) to −φ_{1}(J). These observations allow us to conclude that

\overline{φ_{1}} = φ_{1}, \overline{φ_{2}} = φ_{2}, φ_{1}^{p} = φ_{2},

because there are at most two irreducible Brauer characters of G whose restrictions to H coincide with η_{1}( mod p). Observe that by Lemma 2.4 the character ψ(mod p) is also of type −.

Applying Proposition 2.2 we are convinced that there exists a unique GIR of G with character 2ψ. Furthermore, if n ≥ 2 then, as shown in [51], H has exactly two (nontrivial) irreducible characters of degree dividing 2(p^{n} − 1), namely, η_{1} and η_{2}. If n = 1, one can apply Theorem 4.1. Altogether, we have proved the following result, which supplements Proposition 13.8 from [18].

THEOREM 5.1. Let p ≡ 1(mod 4) be a prime. Then G = Sp_{2n}(p) · 2 has a globally irreducible representation V of dimension 2(p^{n}−1), which is unique if (n, p) ≠ (1, 5). This representation has End_{G}(V) = 𝕂, the quaternion algebra ramified at {p, ∞}.

COROLLARY 5.2. Under the notation of Theorem 5.1 (and after suitable rescaling), every ℤG-lattice Λ in V is even, unimodular, of rank 2(p^{n} − 1), and root-free if p^{n} > 5.

Proof. The fact that Λ is an even unimodular lattice (after suitable rescaling) follows from Proposition 2.10. Consider the set M consisting of all vectors of norm 1 and 2 of Λ and suppose that it is not empty. Then M is a root system in the classical sense. Because ℚH-module V is irreducible, ⟨M⟩_{ℚ} = V and M is an orthogonal sum of ℓ irreducible components, which are isomorphic to each other. Here the integer ℓ divides 2(p^{n}−1), and H permutes transitively these ℓ components. Now if n ≥ 2 then P (H) = \frac{p^{2n}−1}{p−1} > 2(p^{n} − 1). If n = 1, then H = SL_{2}(p) also has no subgroups of such index ℓ. This means that ℓ = 1. Now under the assumption that p^{n} > 5 one sees that M is a root system of type A − D, and so, as a ℚ-irreducible subgroup of Aut(M), H must have a subgroup of index 2(p^{n} −1) or 2p^{n} −1, which is also impossible.

REMARK 5.3. Let Λ denote any of lattices mentioned in Corollary 5.2. If p^{n} = 5, then Λ is the root lattice of type E_{8}. If p^{n} = 13, then Λ is isometric to the Leech lattice. If p^{n} = 17, then Λ is an even unimodular root-free lattice of rank 32, which is not isometric to the Barnes–Wall lattice BW_{32} (see [3]) (because Aut(BW_{32}) = 2_{+}^{1+10} · Ω_{10}^{+}(2) has two conjugacy classes of elements of order 17, while our lattice Λ is invariant under SL_{2}(17) · 2, which has only one conjugacy class of elements of order 17.) Moreover, our lattice Λ and the Barnes-Wall lattice BW_{32} give examples of even unimodular lattices of rank 32 admitting fixedpoint-free automorphisms of order 51. It seems that there exist exactly three (isomorphism classes of) lattices with this property, cf. [2]. This observation allows us to think that our lattice series is new.

6. A Summary of Currently Known Globally Irreducible Representations

We tabulate the currently known globally irreducible representations, and relevant invariants of the corresponding Euclidean lattices. We restrict ourselves to the case where the group is an abelian extension of a quasi-simple group. (However, certain extensions of extraspecial 2-groups, namely, E · Ω_{2n}^{−}(2) and F · Sp_{2n}(2), where E := 2_{−}^{1+2n} and F := 2_{+}^{1+2n} ∗ ℤ_{4}, have also been listed.) Some of these GIR’s have been mentioned in this paper, the others are described elsewhere (cf. [48], [49]). The lines marked by symbol (∗) contain conditions on n and on every prime divisor p of n. 2^{+}𝕊_{n} and 2^{−}𝕊_{n} denote the two double covers of the symmetric group 𝕊_{n}, they are isoclinic to each other. Furthermore, P is the set of all primes ramified in R, and we denote [P] = ∏_{p∈P} p. The word “ℓ-mod.” means that the lattice Λ is ℓ-modular, that is, the lattices Λ and \sqrt{ℓ}Λ^{∗} are isometric (cf. [8], [33]). In particular, if Λ is ℓ-modular, then det Λ = ℓ^{\frac{1}{2} rankΛ}. The fourth column of the table describes the corresponding commuting algebra 𝕂 = End_{ℚ}(V). Here, the symbol {p, ∞} denotes the quaternion algebra ramified at p and ∞. All the lattices in the table are supposed to be stable under a maximal order R of 𝕂. We also tried to name the authors who considered the corresponding lattices.

G  rankΛ  det Λ End_{G}(V)  cf.
2 · O_{8}^{+}(2)  8  1  ℚ  Korkine, Zolotarev
2Co_{1}  24  1  ℚ  Leech, Conway
2 × F_{3}  248  1  ℚ  Thompson-Smith
SL_{2}(3)  4  2-mod.  {2, ∞}  Hurwitz
2𝔸_{5}  8  1  {3, ∞}  Plesken-Pohst
3𝔸_{6} · 2  12  15-mod. ℚ(\sqrt{−15})  Plesken-Nebe
4 · L_{3}(4) · 2  32  ?  ℚ(\sqrt{−5})  Tiep [50]
6 · L_{3}(4) · 2  24  1  {3, ∞}  Leech
6 · 𝔸_{7} · 2  24  2^{12}  ℚ(\sqrt{−6})  Nebe
6 · P SU_{4}(3) · 2  12  3-mod. ℚ(\sqrt{−3})  Coxeter-Todd
2𝔸_{9}, 2𝔸_{10}  96  ?  ℚ(\sqrt{−6})  Tiep
2𝔸_{(2n+1)^{2}}  2^{2n^{2}+2n−1}  1  ℚ  Gow
2𝕊_{2n^{2}}  2^{n^{2}−1}  1  ℚ  Gow
U_{3}(p)  2p(p − 1) p-mod.  {p, ∞}  Elkies, Gross
U_{3}(p^{2})  2p^{2}(p^{2} − 1)  1  {p, ∞}  Elkies, Gross
\{ \begin{matrix}Sp_{4n}(p) \\ p ≡ 3(4) \end{matrix}  p^{2n} − 1  1  ℚ(\sqrt{−p}) Elkies, Gow, Gross

\{ \begin{matrix}Sp_{4n+2}(p) \\ p ≡ 3(4) \end{matrix}  p^{2n+1} − 1 p-mod. ℚ(\sqrt{−p})  Elkies, Gross

\{ \begin{matrix}Sp_{2n}(p^{2}) \\ p ≡ 1(2) \end{matrix}  p^{2n} − 1  1  \{p, ∞\}  Elkies, Gross

\{ \begin{matrix}Sp_{2n}(p) · 2 \\ p ≡ 1(4) \end{matrix}  2(p^{n} − 1)  1  \{p, ∞\}  Tiep (§5)

\{ \begin{matrix}E · Ω_{2n}^{−}(2) \\ n ≡ 0(2) \end{matrix}  2^{n+1}  1  \{2, ∞\} Barnes-Wall, Gross

\{ \begin{matrix}E · Ω_{2n}^{−}(2) \\ n ≡ 1(2) \end{matrix}  2^{n+1}  2-mod.  \{2, ∞\} Barnes-Wall, Gross

\{ \begin{matrix}F · Sp_{2n}(2) \\ n ≡ 0(2) \end{matrix}  2^{n+1}  1  \{2, ∞\}  Barnes-Wall, [49]

\{ \begin{matrix}F · Sp_{2n}(2) \\ n ≡ 1(2) \end{matrix}  2^{n+1}  2-mod.  \{2, ∞\} Barnes-Wall, Gross
SU_{4}(3)  40  3-mod.  {3, ∞}  Tiep [49]
SU_{4}(5)  208  5-mod.  {5, ∞}  Tiep [49]
SU_{5}(2)  20  2-mod.  {2, ∞}  Plesken-Nebe
SU_{5}(3)  120  3-mod.  {3, ∞}  Tiep [49]
SU_{7}(2)  84  2-mod.  {2, ∞}  Tiep [49]
M_{11}, 2M_{12}  20  2-mod. ℚ(\sqrt{−2})  Plesken-Nebe
2M_{22}  20  7-mod. ℚ(\sqrt{−7})  Plesken-Nebe
2J_{2} · 2  24  1  {5, ∞}  Tits
(2J_{2} · 2)^{−}  24  5-mod.  {5, ∞}  Lindsey

G  rankΛ  det Λ  End_{G}(V)  cf.
3J_{3} · 2  72  15-mod.  ℚ(\sqrt{−15}) Conway-Wales
2HS · 2  112  ?  ℚ(\sqrt{−5})  Tiep (§3)
2Ru  56  1  ℚ(\sqrt{−1}) Conway-Wales
6Suz  24  1  ℚ(\sqrt{−3})  Feit
2F_{4}(2)′  52  2-mod.  ℚ(\sqrt{−2})  Tits
2^{+}𝕊_{4n}  2^{2n}  ?  ℚ(\sqrt{−2n})  Tiep [48]
\{ \begin{matrix}2^{+}𝕊_{n}^{(∗)} \\ n ≡ 5(8) \\ p ≡ 3, 5(8) \end{matrix} 2^{(n+1)/2}  [P]^{\frac{1}{2} rankΛ}  quaternion  Tiep [48]

\{ \begin{matrix}2^{+}𝕊_{n}^{(∗)} \\ n ≡ 7(8) \\ p ≡ 5, 7(8) \end{matrix} 2^{(n+1)/2}  1  quaternion  Tiep [48]

\{ \begin{matrix}2^{−}𝕊_{n} \\ n ≡ 2(4) \end{matrix}  2^{n/2}  \{ \begin{matrix}1, & 8|(n − 6) \\ ?, & 8|(n − 2) \end{matrix} ℚ(\sqrt{−n/2})  Tiep [48]

\{ \begin{matrix}2^{−}𝕊_{n}^{(∗)} \\ n ≡ 3(8) \\ p ≡ 3, 5(8) \end{matrix} 2^{(n+1)/2}  [P]^{\frac{1}{2} rankΛ}  quaternion  Tiep [48]

\{ \begin{matrix}2^{−}𝕊_{n}^{(∗)} \\ n ≡ 5(8) \\ p ≡ 5, 7(8) \end{matrix} 2^{(n+1)/2}  1  quaternion  Tiep [48]

\{ \begin{matrix}2𝔸_{n} \\ n ≡ 7(8) \end{matrix}  2^{(n−1)/2}  1  ℚ(\sqrt{−n})  Tiep [48]
\{ \begin{matrix}2𝔸_{n} \\ n ≡ 3(8) \end{matrix}  2^{(n−1)/2}  [P]^{\frac{1}{2} rankΛ}  ℚ(\sqrt{−n})  Tiep [48]

\{ \begin{matrix}2𝔸_{n}^{(∗)} \\ n ≡ 6(8) \\ p ≡ 2, 3(4) \end{matrix}  2^{n/2}  1  quaternion  Tiep [48]

\{ \begin{matrix}2𝔸_{2n^{2}}^{(∗)} \\ p ≡ 3(4) \end{matrix}  2^{n^{2}−1}  1  ℚ  Tiep [48]

PROPOSITION-PROOF 6.1. As we have mentioned above, the GIR’s coming from basic spin representations of 2𝔸_{n} and 2^{±}𝕊_{n} have been classified in [48]. But in that paper the determinant of the arising lattices was determined only in the case of 2𝔸_{8m+7}. Here we compute det Λ for the other cases, Λ an RG-lattice in the corresponding GIR V.

The main tool is provided by Corollary 7.2 and Proposition 8.6 from [18].

1) First let G = 2^{+}𝕊_{n} with n ≡ 5, 7(mod 8) (here and hereafter we omit the condition on prime divisors of n for brevity). Let χ denote the basic spin character of G. Then χ(1) = 2^{(n−1)/2}, ℚ(χ) = ℚ, ind(χ) = −1. If p |̸n, then χ(mod p) ∈ IBr_{p}(G). In particular, by Note 4.3 [18], all the primes ramified in 𝕂 are divisors of n. In order to compute det Λ, one needs to consider only the ramified primes p. For such a p, as is shown in [48], χ(mod p) = φ + φ^{(p)} for a certain φ ∈ IBr_{p}(G).

Moreover, φ(x) = \sqrt{(−1)^{(n+1)/2}(n − 1)/2}, if x ∈ G has cycle type (n − 1)(1) in 𝕊_{n}.

1a) Let n ≡ 5(mod 8). Then φ(x) ∉ ℝ, which implies that φ ≠ φ̄. Thus V_{℘} (as defined in (2)) is an absolutely irreducible Hermitian representation over k_{℘} = 𝔽_{p^{2}}. By Proposition 8.6 [18], det Λ =

(∏_{p∈P} p)^{\frac{1}{2} rankΛ}, where P denotes the set of the primes ramified in R.

1b) Let n ≡ 7(mod 8). Then φ(x) = \overline{φ(x)} ≠ 0, which implies that φ = φ̄. From the equality ind(χ) = −1 it follows that φ is of type −, i.e., V_{℘} is an absolutely irreducible symplectic representation over k_{℘} = 𝔽_{p^{2}}. By Proposition 8.6 [18], det Λ = 1.

2) Next let G = 2^{−}𝕊_{n} with n ≡ 3, 5(mod 8). Let χ denote the basic spin character of G. Then χ(1) = 2^{(n−1)/2}, ℚ(χ) = ℚ, ind(χ) = −1. As above, every ramified prime p divides n. For such a p, as is shown in [48], χ(mod p) = φ + φ^{(p)} for a certain φ ∈ IBr_{p}(G). Moreover,

φ(x) = \sqrt{(−1)^{(n−1)/2}(n − 1)/2}, if x ∈ G has cycle type (n − 1)(1) in 𝕊_{n}.

2a) Let n ≡ 3(mod 8). Then φ(x) ∉ ℝ, which implies that φ ≠ φ̄. Thus V_{℘} is an absolutely irreducible Hermitian representation over

k_{℘} = 𝔽_{p^{2}}. By Proposition 8.6 [18], det Λ = (∏_{p∈P} p)^{\frac{1}{2} rankΛ}, where P denotes the set of the primes ramified in R.

2b) Let n ≡ 5(mod 8). Then φ(x) = \overline{φ(x)} ≠ 0, which implies that φ = φ̄. From the equality ind(χ) = −1 it follows that φ is of type −, i.e., V_{℘} is an absolutely irreducible symplectic representation over k_{℘} = 𝔽_{p^{2}}. By Proposition 8.6 [18], det Λ = 1.

3) Here G = 2𝔸_{n} with n ≡ 3, 7(mod 8). Let χ denote any of the basic spin characters of G. Then χ(1) = 2^{(n−3)/2}, ℚ(χ) = ℚ(\sqrt{−n}). Every ramified prime p divides n. For such a p, as is shown in [16], χ(mod p) ∈ IBr_{p}(G). Moreover, χ(mod p) is of type + if n ≡ 3(mod 8), and − if n ≡ 7(mod 8) (cf. [48]). Hence det Λ = 1 in case n ≡

7(mod 8). In case n ≡ 3(mod 8), det Λ = (∏_{p∈P} p)^{\frac{1}{2} rankΛ}, where P denotes the set of the primes ramified in R (that is, P is the set of primes which enter n with odd multiplicity).

4) Let H = 2𝔸_{n} with n ≡ 6(mod 8). Let α denote the basic spin character of H. Then α(1) = 2^{(n−2)/2}, ℚ(α) = ℚ, ind(α) = −1. Since α(mod 2) ∈ IBr_{2}(H) (cf. [48]), 2 is unramified in R. Similarly, p is unramified in R if (p, n) = 1. For any ramified prime p, as is shown in [48], α(mod p) = φ + φ^{(p)} for a certain φ ∈ IBr_{p}(G). Moreover,

φ(x) = \sqrt{(−1)^{(n−2)/2}(n − 2)/2}, if x ∈ H has cycle type (n − 2)(1)(1) in 𝔸_{n}. Now the condition n ≡ 6(mod 8) implies φ(x) ∈ ℝ^{∗}, φ = φ̄, and the equality ind(α) = −1 implies that φ is of symplectic type. Applying Proposition 8.6 [18], we obtain det Λ = 1.

5) Finally, let G = 2^{−}𝕊_{n} with n ≡ 6( mod 8). Let χ denote any of the basic spin characters of G. Then χ(1) = 2^{(n−2)/2}, ℚ(χ) = ℚ(\sqrt{−n/2}). Every ramified prime p divides n/2. For such a p, as is mentioned in [48], χ(mod p) ∈ IBr_{p}(G). Let us determine the type of this Brauer character. Observe that χ( mod p) vanishes at all p′-elements contained in G \ H, where H = 2𝔸_{n}. Moreover, α = χ|_{H} is the basic spin character of H. Hence χ(mod p) is real-valued: it is of type ϵ, ϵ = ±. The arguments in 4) and Lemma 2.4 ensure that ϵ = −. Consequently, det Λ = 1.

6) The same arguments are applicable to G = 2^{−}𝕊_{n} with n ≡ 2( mod 8) and G = 2^{+}𝕊_{n} with 4|n; but the results are not so complete. In the

former case one can show that det Λ = (∏_{p∈P} p)^{\frac{1}{2} rankΛ}, where P is either the set of all ramified primes, or the set of all ramified primes but 2.

In the latter case let χ denote any of the basic spin characters of G. Then for any odd prime p dividing n one can show that χ( mod p) is an absolutely irreducible Brauer character of type ϵ, where ϵ = (−1)^{1+n/4}.

Thus det Λ = (∏_{p∈P} p)^{\frac{1}{2} rankΛ}, where P = ∅ or \{2\}, if 8|n. If n ≡ 4(mod 8), then P is either the set of all ramified primes or the set of all odd primes which ramify in R. In particular, if n = 2^{2a+3}(4b + 3) for a, b ≥ 0, then 2 is unramified, and any RG-lattice is unimodular.

EXAMPLE 6.2. 1) 2^{+}𝕊_{5} stabilizes a 5-modular lattice of rank 8 (cf. [29], p. 25). 2^{+}𝕊_{13} stabilizes a 13-modular lattice of rank 128. 2^{+}𝕊_{7} stabilizes the root lattice 2E_{8}.

2) 2^{−}𝕊_{11} stabilizes a 11-modular lattice of rank 64. 2^{−}𝕊_{13} and 2^{−}𝕊_{14} stabilize an even unimodular lattice of rank 128.

3) 2𝔸_{7} stabilizes the root lattice E_{8}. 2𝔸_{11} stabilizes an 11-modular lattice of rank 32.

REMARK 6.3. The table still contains several question marks; namely in the lines of 4 · L_{3}(4) · 2, 2𝔸_{10}, 2HS · 2, 2^{+}𝕊_{4n} and 2^{−}𝕊_{8n+2}. In these cases, we have not been able to compute det Λ by the reason that 2 ramifies in R. For the first three cases we conjecture that det Λ is 2^{16} · 5^{16}, 2^{48} · 3^{48}, 2^{56}, respectively.

Acknowledgements

I am very grateful to Professor G. O. Michler and his colleagues at the Institute for Experimental Mathematics for stimulating conversations and their generous hospitality. My sincere thanks go to Professors E. Bayer, A. M. Cohen, B. H. Gross, H. Koch, W. Plesken, H.-G. Quebbemann, R. Scharlau, J.-P. Serre and J. Tits for valuable comments on the paper. I would also like to thank the referees for their suggestions which greatly improve the exposition of the paper. Finally, the financial support of the Alexander von Humboldt Foundation is gratefully acknowledged.

References

1. Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A. and Wilson, R. A.: An ATLAS of Finite Groups, Clarendon Press, Oxford, 1985.

2. Bayer-Fluckiger, E.: Definite unimodular lattices having an automorphism of given characteristic polynomial, Comment. Math. Helvetici 59 (1984), 509 − 538.

3. Barnes, E. S. and Wall, G. E.: Some extreme forms defined in terms of abelian groups, J. Austral. Math. Soc. 1 (1959), 47 − 63.

4. Burkhardt, R.: Die Zerlegungsmatrizen der Gruppen P SL(2, p^{f}), J. Algebra 40 (1976), 75 − 96.

5. Burkhardt, R.: Über die Zerlegungszahlen der Suzukigruppen Sz(q), J. Algebra 59 (1979), 421 − 433.

6. Conway, J. H.: A characterization of Leech’s lattice, Invent. Math. 7 (1969), 137 − 142.

7. Conway, J. H.: A group of order 8315553613086720000, Bull. London Math. Soc. 1 (1969), 79 − 88.

8. Conway, J. H. and Sloane, N. J. A.: On lattices equivalent to their duals, J. Number Theory 48 (1994), 373 − 382.

9. Dornhoff, L.: Group Representation Theory, Dekker, New York, 1971.

10. Dummigan, N.: Symplectic group lattices as Mordell-Weil sublattices, (Preprint).

11. Elkies, N.: On Mordell-Weil lattices, in Arbeitstagung 1990, Max-Planck-Institut f. Mathematik, Bonn.

12. Feit, W.: On integral representations of finite groups, Proc. London Math. Soc. (3) 29 (1974), 633 − 683.

13. Feit, W.: Some lattices over Q(\sqrt{−3}), J. Algebra 52 (1978), 248 − 263.

14. Feit, W.: The Representation Theory of Finite Groups, North-Holland Publishing Company, Amsterdam et al, 1982.

15. Gow, R.: Even unimodular lattices associated with the Weil representation of the finite symplectic group, J. Algebra 122 (1989), 510 − 519.

16. Gow, R.: Unimodular integral lattices associated with the basic spin representations of 2A_{n} and 2S_{n}, Bull. London Math. Soc. 21 (1989), 257 − 262.

17. Gow, R. and Willems, W.: On the geometry of some simple modules in characteristic 2, Preprint 10 (1993), Institute for Experimental Mathematics, University of Essen.

18. Gross, B. H.: Group representations and lattices, J. Amer. Math. Soc. 3 (1990), 929 − 960.

19. Gross, B. H.: Groups over ℤ, Invent. Math. 124 (1996), 263 − 279.

20. Hiss, G. and Lux, K.: The Brauer characters of the Hall–Janko group, Commun. Algebra 16 (1988), 357 − 398.

21. Hiss, G. and Lux, K.: Brauer Trees of Sporadic Groups, Clarendon Press, Oxford, 1989.

22. Humphreys, J. F.: The projective characters of the Mathieu group M_{22}, J. Algebra 76 (1982), 1 − 24.

23. Isaacs, I. M.: Character Theory of Finite Groups, Academic Press, New York, 1976.

24. James, G. D.: The modular characters of the Mathieu groups, J. Algebra 27 (1973), 57 − 111.

25. Jansen, C., Lux, K., Parker, R. A. and Wilson, R. A.: An Atlas of Brauer Characters, (in preparation).

26. Landázuri, V. and Seitz, G.: On the minimal degrees of projective representations of the finite Chevalley groups, J. Algebra 32 (1974), 418 − 443.

27. Lempken, W. and Staszewski, R.: A construction of ˆ3McL and some representation theory in characteristic 5, Linear Algebra Appl. 192 (1993), 205 − 234.

28. Lindsey, J. H.: A new lattice for the Hall-Janko group, Proc. Amer. Math. Soc. 103 (1988), 703 − 709.

29. Nebe, G.: Endliche Rationale Matrixgruppen vom Grad 24, Aachener Beiträge zur Mathematik, Bd. 12, 1995.

30. Nebe, G. and Plesken, W.: Finite Rational Matrix Groups, Mem. Amer. Math. Soc. 116 (1995), no. 556.

31. Pierse, R.: Associative Algebras, Springer-Verlag, New York, 1982.

32. Plesken, W.: Some applications of representation theory, Progress in Math. 95 (1991), 477 − 496.

33. Quebbemann, H.-G.: Modular lattices in Euclidean spaces, (to appear in J. Number Theory).

34. Reiner, I.: Maximal Orders, Academic Press, 1975.

35. Ryba, A. J. E.: Calculations of the 7-modular characters of the Held group, J. Algebra 117 (1988), 240 − 255.

36. Shioda, T.: Mordell-Weil lattices and sphere packings, Amer. J. Math. 113 (1991), 931 − 948.

37. Smith, P. E.: On Certain Finite Simple Groups, Ph. D. Thesis, Cambridge, 1975.

38. Smith, P. E.: A simple subgroup of M? and E_{8}(3), Bull. London Math. Soc. 8 (1976), no. 9, 161 − 165.

39. Srinivasan, B.: On the modular characters of the special linear group SL(2, p^{n}), Proc. London Math. Soc. (3) 14 (1964), 101 − 114.

40. Srinivasan, B.: The characters of the finite symplectic group Sp(4, q), Trans. Amer. Math. Soc. 131 (1968), 488 − 525.

41. Thompson, J. G.: Finite groups and even lattices, J. Algebra 38 (1976), 523 − 524.

42. Thompson, J. G.: A simple subgroup of E_{8}(3), in Finite Groups Symposium, N. Iwahori, Japan Soc. for Promotion of Science, 1976, pp. 113 − 116.

43. Thompson, J. G.: Bilinear forms in characteristic p and the Frobenius–Schur indicator, in Group Theory, Beijing 1984, Lecture Notes in Math. no. 1185, pp. 221 − 230.

44. Pham Huu Tiep: Weil representations of finite symplectic groups, and Gow lattices, Mat. Sb. 182 (1991), no. 8, 1161 − 1183; English transl. in Math. USSR-Sb. 73 (1992), no. 2, 535 − 555.

45. Pham Huu Tiep: Automorphism groups of some Mordell-Weil lattices, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 3, 509 − 537; English transl. in Russian Acad. Sci. Izv. Math. 40 (1993), no. 3, 477 − 501.

46. Pham Huu Tiep: Basic spin representations of alternating groups, Gow lattices, and Barnes–Wall lattices, Mat. Sb. 183 (1992), no. 11, 99−116; English transl. in Russian Acad. Sci. Sb. Math. 77 (1994), no. 2, 351 − 365.

47. Pham Huu Tiep: Globally irreducible representations of the finite symplectic group Sp_{4}(q), Commun. Algebra 22 (1994), 6439 − 6457.

48. Pham Huu Tiep: Basic spin representations of 2𝕊_{n} and 2𝔸_{n} as globally irreducible representations, Archiv Math. 64 (1995), 103 − 112.

49. Pham Huu Tiep: Weil representations as globally irreducible representations, (to appear in Math. Nachr.).

50. Pham Huu Tiep: Globally irreducible representations of SL_{3}(q) and SU_{3}(q), (submitted to Israel J. Math.).

51. Pham Huu Tiep and Zalesskii, A. E.: Minimal characters of the finite classical groups, Commun. Algebra 24 (1996), no. 6.

52. Tits, J.: Quaternions over ℚ(\sqrt{5}), Leech’s lattice and the sporadic group of Hall-Janko, J. Algebra 63 (1980), 56 − 75.

53. Vignéras, M.-F.: Arithmétique des algèbres de quaternions, Lecture Notes in Math., no. 800, Springer, Berlin, 1980.

54. Woldar, A. J.: On the 5-decomposition matrix for McLaughlin’s sporadic simple group, Commun. Algebra 14 (1986), 277 − 291.