Invariants of orthogonal G-modules from the character table.

Gabriele Nebe

Abstract: Character theoretic methods using Clifford algebras are developed to describe the quadratic forms on a vector space V that are invariant under a finite subgroup G of GL(V) such that the order of the commutator factor group of G is odd.

1 Introduction

Let G be a finite group. Any ℚG-module V is uniquely determined by its character χ_{V}. So in principle χ_{V} also determines the G-invariant quadratic forms on V. However, there is not much known how to calculate rational invariants of these quadratic forms without describing the action of G on V explicitly. Of course, the G-invariant form q on V is not unique, since for all a ∈ ℚ, the form aq is also G-invariant, and there are more G-invariant forms, if the V is not irreducible over the reals. So one can only hope to calculate say the determinant of q, which is defined as the determinant of a Gram matrix of the corresponding bilinear form modulo squares, if dim(V) is even.

Invariants for a non-degenerate quadratic space φ := (V, q) over an arbitrary field K can be read off from the Clifford algebra C(φ). This is a ℤ/2ℤ-graded algebra functorially attached to φ and it determines the two most important invariants of the quadratic space φ: its determinant and its Clifford invariant (see Theorem 2.1). According to [Hasse 1924], if V is a vector space over a number field K, then these two invariants together with the dimension of V and the signatures of φ at all real places of K describe the K-isometry class of the non-degenerate quadratic space φ over K completely.

If G acts on φ = (V, q) as isometries, then G acts on C(φ) as algebra automorphisms respecting the grading. If dim(V) is even, then C(φ) =: c(φ) is central simple and if dim(V) is odd, then the even part c(φ) := C_{0}(φ) of the Clifford algebra is a central simple algebra. So the simple c(φ)-module W becomes a module for some projective representation of G (in the sense of Schur) of which the character χ_{W} (of a covering group of G) is a certain square root of a character constructed from χ_{V}. An irreducible constituent that occurs in χ_{W} with odd multiplicity gives information on either the determinant of φ, if dim(V) is even, or the Clifford invariant of φ, if dim(V) is odd (see Corollary 3.6). Applications are given in the examples in Section 4.

Related ideas have been used by A. Turull in [Turull 1992], where he determines the Schur index of χ_{W} using the explicit knowledge of (V, q) for the (n−1)-dimensional S_{n}-module V.

This paper summarizes one idea of my Habilitationsschrift [Nebe 1999]. There I develop also other methods to calculate the isometry class of φ using constructions of V as a constituent of a tensor product or of an induced module (see also [Nebe 2000]).

I thank Prof. Carl Riehm for valuable questions and corrections on this paper.

2 Clifford algebras

Let φ := (V, q) be a quadratic space of dimension n := dim(V) over a field K. Then the Clifford algebra

C(φ) := T (V)/I(φ)

is the quotient of the tensor algebra T (V) := ⊕_{i=0}^{∞} ⊗^{i} V by the two-sided ideal I(φ) generated by v ⊗ v − q(v) · 1 with v ∈ V (see [Scharlau 1985, Chapter 9]). C(φ) is a 2^{n}-dimensional K-algebra. It contains the even Clifford algebra

C_{0}(φ) := ⟨v_{1} … v_{s} | v_{i} ∈ V, s even ⟩ ≤ C(φ)

as a subalgebra of dimension 2^{n−1} = dim_{K}(C_{0}(φ)). Let C_{1}(φ) := ⟨v_{1} … v_{s} | v_{i} ∈ V, s odd ⟩. If (v_{1}, …, v_{n}) is a basis of V, then (v_{i_{1}} … v_{i_{s}} | 1 ≤ i_{1} < … < i_{s} ≤ n) is a basis of C(φ). In particular, V is embedded in C(φ).

If char(K) ≠ 2, then let B_{q} be the bilinear form associated to q, defined by

B_{q}(v, w) := \frac{1}{2}(q(v + w) − q(v) − q(w))

for all v, w ∈ V. Then the determinant det(φ) is the determinant of a Gram matrix of B_{q} modulo squares

det(φ) := det(B_{q}(v_{i}, v_{j}))_{i,j=1}^{n}(K^{∗})^{2} ∈ K/(K^{∗})^{2}

and the discriminant of φ is d_{±}(φ) = (−1)^{\binom{n}{2}} det(φ) ∈ K/(K^{∗})^{2}. Assume that char(K) ≠ 2 and φ is non-degenerate, i.e. d_{±}(φ) ≠ 0. Then one has

Theorem 2.1 ([Scharlau 1985, Theorem (9.2.10)])

(i) If n is even, then C(φ) is a central simple K-algebra and Z(C_{0}(φ)) is isomorphic to K[X]/(X^{2} − d_{±}(φ)).

(ii) If n is odd, then C_{0}(φ) is a central simple K-algebra and Z(C(φ)) is isomorphic to K[X]/(X^{2} − d_{±}(φ)).

An analogous theorem holds if char(K) = 2, see [Knus 1991, Theorem (2.2.3)]. Let

c(φ) := \{ \begin{matrix}C(φ) & dim(V) even \\ C_{0}(φ) & dim(V) odd. \end{matrix}

The class [c(φ)] ∈ Br(K) of the central simple K-algebra c(φ) in the Brauer group Br(K) of K is called the Clifford invariant of φ.

Remark 2.2 (i) Let W be the simple right c(φ)-module, D := End_{c(φ)}(W) and W^{∗} := Hom_{D}(W, D) the simple left c(φ)-module. Then c(φ) ≅ W^{∗} ⊗_{D} W as c(φ)-bimodule.

(ii) Because of the universal property of C(φ), the identity embedding of V ⊂ C(φ) extends to a unique algebra antiautomorphism of C(φ), the canonical involution . The involution satisfies \overline{v_{1} … v_{s}} = v_{s} … v_{1} for all v_{i} ∈ V and induces an algebra antiautomorphism on the central simple K-algebra c(φ). Via this involution W^{∗} becomes a right c(φ)-module isomorphic to W.

2.1 The orthogonal group.

For a non-degenerate quadratic space φ = (V, q) the orthogonal group O(φ) is defined as

O(φ) := {f ∈ GL(V) | q(f(v)) = q(v) for all v ∈ V }.

Using the universal property of C(φ) one easily sees that the linear action of O(φ) on V extends uniquely to K-algebra automorphisms of the Clifford algebra C(φ). This action of O(φ) on C(φ) respects the grading.

The Clifford group is

Γ(φ) := {s ∈ C_{0}(φ)^{∗} ∪ (C(φ)^{∗} ∩ C_{1}(φ)) | sV s^{−1} = V }

(see [Scharlau 1985, Section 9.3], [Knus 1991, Section IV.6]). For s ∈ Γ(φ) let γ(s) = 1 if s ∈ C_{0}(φ) and γ(s) = −1 if s ∈ C_{1}(φ). Define

α(s) : V → V; v ↦ γ(s)s^{−1}vs.

Then α(Γ(φ)) ⊂ O(φ). If s ∈ V ⊂ C_{1}(φ) satisfies q(s) ≠ 0, then s ∈ Γ(φ) and α(s) ∈ O(φ) is the reflection along the anisotropic vector s. If char(K) ≠ 2, then the reflections along anisotropic vectors in V generate O(φ). Therefore α(Γ(φ)) = O(φ). If char(K) = 2 then α is also surjective [Knus 1991, p. 228]. The kernel of α is K^{∗} ⊂ C_{0}(φ)^{∗} and one has the following exact sequence

1 → K^{∗} → Γ(φ) → O(φ) → 1

which gives rise to a projective representation P : O(φ) → Γ(φ) mapping the reflection along the anisotropic vector s ∈ V to s ∈ Γ(φ). By [Scharlau 1985, Lemma 9.3.2], ss̄ ∈ K^{∗} ⊆ C_{0}(φ) for all s ∈ Γ(φ), where is the canonical involution of c(φ) (cf. Remark 2.2). This gives rise to a group homomorphism

norm : O(φ) → K^{∗}/(K^{∗})^{2}, norm(g) := P (g) · \overline{P (g)} · (K^{∗})^{2}

called the Spinor norm (cf. [Scharlau 1985, Definition 9.3.4]). Let

SΓ(φ) := {s ∈ Γ(φ) ∩ C_{0}(φ) | s̄ = s^{−1}}.

Proposition 2.3 The group SΓ(φ) acts on c(φ) by conjugation: c(φ) × SΓ(φ) → c(φ), (x, s) ↦ s^{−1}xs. With the notations of Remark 2.2 one has

c(φ) ≅ W ⊗_{D} W

as SΓ(φ)-modules.

3 Clifford algebras as G-algebras.

Let φ = (V, q) be a non-degenerate quadratic space and G be a subgroup of the orthogonal group O(φ). Then φ is also called an orthogonal KG-module. Theoretical concepts for Clifford algebras as G-algebras are given in [Fröhlich 1972]. Here practical methods to obtain information on C(φ) from the character χ = χ_{V} of the G-module V are developed.

Since G ≤ O(φ), the action of O(φ) on C(φ) restricts to a linear representation Δ_{C(φ)} of G on the Clifford algebra that respects the grading:

Remark 3.1 The character of the KG-module C(φ) respectively C_{0}(φ) is

χ̃ := ∑_{i=0}^{n} Λ^{i}(χ) respectively χ̃_{0} := ∑_{i=0,i even}^{n} Λ^{i}(χ)

where Λ^{i}(χ) is the i-th exterior power of χ.

Note that

χ̃(g) = (−1)^{n}p_{g}(−1) for all g ∈ G

where p_{g} is the characteristic polynomial of g on V. With this trick one can calculate χ̃(g) (and χ̃_{0}(g)) with the help of GAP, cf. [GAP], by restricting χ to the subgroup ⟨g⟩ ≤ G, for any group G whose character table and powermap is known.

Assumption. From now on we assume that the order of the commutator factor group G/G′ is odd.

Lemma 3.2 There are a_{g} ∈ K^{∗}( for all g ∈ G) such that P_{0}(g) := a_{g}P (g) satisfies P_{0}(g) ⊆ C_{0}(φ), P_{0}(g)\overline{P_{0}(g)} = 1, and P_{0}⊗P_{0} : G → GL(c(φ)) is a linear representation equivalent to Δ_{c(φ)}.

\underline{Proof.} The mapping P : G → C(φ)^{∗}, g ↦ P (g) is a projective representation of G. Hence P (g)P (h) = a(g, h)P (gh) for some a(g, h) ∈ K^{∗} and for all h, g ∈ G. Since K^{∗} ⊆ C_{0}(φ), the grading of the Clifford algebra defines a group homomorphism G → ℤ/2ℤ, g ↦ deg(P (g)). Since |G/G′| is odd, this group homomorphism is trivial, hence P (G) ⊆ C_{0}(φ). In particular P (G) ⊆ c(φ) and P is a projective representation of G on the simple c(φ)-module W. Also the Spinor norm norm : G → K^{∗}/(K^{∗})^{2} is a homomorphism from G to an abelian 2-group and hence trivial. Rescaling P (g) with elements in K^{∗} we may therefore assume that P (g)\overline{P (g)} = 1, i.e. P (g) ∈ SΓ(φ) for all g ∈ G.

Therefore, by Proposition 2.3, the G-module c(φ) is isomorphic to the tensor square W ⊗_{D} W of the projective G-module W. In particular the projective representation P ⊗_{D} P is equivalent (as a projective representation) to the linear representation Δ_{c(φ)} of G on c(φ). This means that there is T ∈ GL(c(φ)) and a mapping α : G → K^{∗} such that T P (g) ⊗ P (g)T^{−1} = α(g)Δ_{c(φ)}(g) for all g ∈ G. Since P is a projective representation and Δ_{c(φ)} is linear, α : G → K^{∗}/(K^{∗})^{2} is a homomorphism. Again, since |G/G′| is odd, we have α(g) = β(g)^{2} ∈ (K^{∗})^{2} for all g ∈ G and P_{0}(g) := β(g)^{−1}P (g) has the desired properties. □

Corollary 3.3 (cf. [Gagola and Garrison 1982, Theorem 1.2, Corollary 4.3]) Assume that char(K) ≠ 2 and let g ∈ G be an element of order 2 and e the dimension of the −1-eigenspace of g in V. Then P_{0}(g)^{2} = (−1)^{\binom{e}{2}}id.

\underline{Proof.} Let v_{1}, …, v_{e} be an orthogonal basis of the −1-eigenspace of g on V. Then P_{0}(g) = a_{g}v_{1} … v_{e} and \overline{P_{0}(g)} = a_{g}v_{e} … v_{1} = (−1)^{\binom{e}{2}}P_{0}(g). Since P_{0}(g)\overline{P_{0}(g)} = id, one has P_{0}(g)^{2} = (−1)^{\binom{e}{2}}id. □

If c(φ) ≅ D^{a×a} for some central K-division algebra D, then the simple c(φ)- module W is isomorphic to D^{a}. Over the algebraic closure of K, the c(φ)-module W is isomorphic to the sum of m copies of a simple module, where m is the index of D (dim_{K}(D) = m^{2}).

We now fix a covering group u : G̃ → G of G such that P_{0} is equivalent to a linear representation of G̃. Let W be the simple c(φ)-module and m the index of End_{c(φ)}(W).

Corollary 3.4 Let mχ_{W} be the character of a linear KG̃-module that is equivalent to W over the algebraic closure of K. Regarding χ as a character of G̃ one has

χ_{W} ⊗ χ_{W} = \{ \begin{matrix}χ̃ & n even \\ χ̃_{0} & n odd. \end{matrix}

For the next theorem we additionally assume that K is a number field. By [Scharlau 1985, Lemma 9.2.8] c(φ) is a tensor product of quaternion algebras. Since K is a number field, this implies that c(φ) is a matrix ring over a quaternion algebra (see for instance [Reiner 1975, Theorem (32.9)]) and C_{0}(φ) ≅ D^{a×a} where D = L := Z(C_{0}(φ)) or D is a quaternion division algebra over L.

Theorem 3.5 With the notations above let m be the Schur index of D, W the simple C_{0}(φ)-module and mχ_{W} the corresponding character of G̃. Assume that there is an absolutely irreducible character ψ of G̃ occurring with odd multiplicity in χ_{W}.

(a) If n is even and L = Z(C_{0}(φ)) is a field then L is a subfield of the character field K(ψ).

(b) Assume that n is odd. If the Schur index of ψ is odd, then K(ψ) splits D. Otherwise let U be the irreducible KG̃-module whose character contains ψ. Then D ⊂ End_{KG̃}(U).

\underline{Proof.} By Lemma 3.2 P (G) is already contained in C_{0}(φ) and therefore End_{C_{0}(φ)}(W) ⊆ EndG̃(W).

In both cases C_{0}(φ) ≅ D^{a×a} and dim_{K}(C_{0}(φ)) = a^{2}m^{2}[L : K] = 2^{n−1} is a power of 2. Let x be the multiplicity of ψ in χ_{W}, U the irreducible K-module whose character contains ψ and D_{U} := End_{KG̃}(U). Then D_{U} is a skew field with center K(ψ) and of index, say, m_{U}. Let U′ be the U-homogeneous component in W_{|G̃}. Then End_{KG̃}(U′) ≅ D_{U}^{y×y} for some y ∈ ℕ. Since the multiplicity of ψ in mχ_{W} and the multiplicity of ψ in χ_{U′} are equal, one has

⋆ mx = ym_{U}.

Since D has no zero divisors, D embeds into End_{KG̃}(U′) and hence

A := D_{U}^{op} ⊗_{K(ψ)} (D ⊗_{K} K(ψ)) → D_{U}^{op} ⊗_{K(ψ)} D_{U}^{y×y} ≅ K(ψ)^{m_{U}^{2} y×m_{U}^{2} y} =: B

where D_{U}^{op} denotes the opposite algebra of D_{U}. If K(ψ) ⊗_{K} L is a field then let ϵ := [L : K] ∈ {1, 2}. Then A is a central simple K(ψ) ⊗_{K} L-algebra isomorphic to D̃^{k×k} for some central K(ψ) ⊗_{K} L-division algebra D̃ of index, say, m̃. If K(ψ) ⊗_{K} L is not a field then let ϵ := 1. Then A is a direct sum of two isomorphic central simple K(ψ)-algebras isomorphic to D̃^{k×k} for some central K(ψ)-division algebra D̃ of index, say, m̃.

In both cases the dimension of A over its center is

(⋆⋆) m_{U}^{2} · m^{2} = m̃^{2} · k^{2}

and the K(ψ)-dimension of a simple A-module is ϵ · m̃^{2} · k and divides the K(ψ)- dimension of the simple B-module, which is m_{U}^{2} · y

ϵ · m̃^{2} · k divides m_{U}^{2} · y.

We claim that m̃ is odd and ϵ = 1. Since K is a number field, m is either 1 or 2. If m = 1, then m_{U} and y are odd by ⋆ (recall that x is odd) and hence also m̃ and ϵ are odd. Assume that m = 2. Then ⋆ implies that either m_{U} is even and y · \frac{m_{U}}{2} is odd, or y is even and m_{U} · \frac{y}{2} is odd. Assume that 2 | m̃. If m_{U} is even, then 2^{3} divides m̃^{2} · k and if m_{U} is odd, then 2^{2} divides m̃^{2} · k. But this power of 2 does not divide m_{U}^{2} · y in both cases, which is a contradiction. Therefore m̃ is odd and k is even. If m_{U} is even, then 4 divides k by (⋆⋆) and therefore ϵ is odd. If m_{U} is odd, then also ϵ = 1 since k is even and \frac{y}{2} is odd. Therefore the claim follows. In particular ϵ = 1 and hence L is a subfield of K(ψ) which proves (a). Now we prove (b). Since n is odd, L = K and ([D_{U}]^{−1} · [D ⊗_{K} K(ψ)]) has odd order in the Brauer group of K(ψ) because

([D_{U}]^{−1} · [D ⊗_{K} K(ψ)])m̃ = 1 ∈ Br(K(ψ))

Therefore the local index m_{℘}(D_{U}) is even, if and only if the local index m_{℘}(D⊗_{K} K(ψ)) is 2, for all (infinite and finite) places ℘ of K(ψ). Hence D ⊗_{K} K(ψ) embeds into D_{U} (cf. [Reiner 1975, Exercise (29.7)]). □

In the applications absolutely irreducible orthogonal G-modules (V, q) over totally real number fields K are of special interest. Then q is (positive or negative) definite. If n is even, then the discriminant of q is negative, if n ≡ 2 (mod 4) and positive, if 4 | n.

Corollary 3.6 In addition to the assumptions of the theorem let K be a totally real number field and assume that φ is definite.

(a) Let n be even. If [K(ψ) : K] is odd or n ≡ 0 (mod 4) and all intermediate fields K(ψ) ⊃ L ⊃ K of degree [L : K] = 2 are complex fields, then the discriminant d_{±}(φ) = 1. If n ≡ 2 (mod 4), then K(ψ)/K has a totally complex intermediate field L with [L : K] = 2. One of these fields is isomorphic K[\sqrt{d_{±}(φ)}].

(b) Assume that n is odd. If ψ has Schur index 1, then the Clifford invariant [c(φ)]

satisfies

[c(φ) ⊗_{K} K(ψ)] = [K(ψ)] ∈ Br_{2}(K(ψ)).

If ψ has Schur index 2 then [K(ψ) ⊗_{K} c(φ)] = [End_{K(ψ)G̃}(U)] ∈ Br_{2}(K(ψ)) for the irreducible K(ψ)G-module U with character 2ψ.

4 Examples.

We now apply the methods presented before to some irreducible representations of finite quasi simple groups. The notations are taken from [Conway et al. 1985].

1) Let G ≅ 2.O_{8}^{+}(2). Then G is perfect and its universal covering group is G̃ ≅ 2^{2}.O_{8}^{+}(2). Let V be the 8-dimensional faithful ℚG-module with character χ and q a non zero G-invariant quadratic form on V. If φ := (V, q), then dim(c(φ)) = 2^{8} and χ̃ = χ_{W} ⊗ χ_{W} for a 16-dimensional G̃-module W. One calculates that χ_{W} = χ_{8} + χ′_{8} is the sum of the two irreducible characters χ_{8}, χ′_{8} ≠ χ which belong to absolutely irreducible rational modules of degree 8 of G̃. Therefore d_{±}(φ) = 1 and also [c(φ)] = [ℚ]. Of course, that d_{±}(φ) = 1 is well known and can also be seen by inspection of the modular constituents of V (cf. [Jansen et al. 1995]).

2) Let G ≅ M^{c}L and φ := (V, q) a 22-dimensional orthogonal ℚG-module with character χ. The universal covering group of G is 3.G. Therefore P_{0} : G → c(φ) can be chosen to be linear. There is a unique character χ_{W} of G satisfying χ_{W} ⊗ χ_{W} = χ̃. In the notation of [Conway et al. 1985] one has χ_{W} = 2(χ_{1} + χ_{2} + χ_{3}) + χ_{5} + χ_{6}. Now the character field ℚ[χ_{5}] = ℚ[χ_{6}] = ℚ[\sqrt{−15}]. Since dim(V) ≡ 2 (mod 4), Corollary 3.6 yields d_{±}(φ) = −15.

3) Let G ≅ S_{6}(3) and χ the irreducible character of degree 78 with orthogonal ℚGmodule φ := (V, q). The universal covering group of G is G̃ ≅ 2.S_{6}(3). Let χ_{W} be the character of G̃ on the simple c(φ)-module. If g ∈ G is an element of order 2 in class 2B in the notation of [Conway et al. 1985], then −g has a 42-dimensional fixed space on V. Therefore χ_{W} is a faithful character of G̃ by Corollary 3.3. With GAP one finds that there is only one faithful character χ_{W} of G̃ satisfying χ_{W} ⊗ χ_{W} = χ̃. The character χ_{W} contains the two complex conjugate irreducible characters ψ_{1} and ψ_{2} of degree 13 with multiplicity 1683. Since dim(V) ≡ 2 (mod 4) and ℚ[ψ_{1}] = ℚ[\sqrt{−3}] Corollary 3.6 yields d_{±}(φ) = −3.

4) The applications are not restricted to the characteristic 0 case. Let V be the 4dimensional 𝔽_{2}A_{6}-module. If V admits a non-degenerate A_{6}-invariant quadratic form q, then there is a projective representation A_{6} → C_{0}((V, q))^{∗} yielding an irreducible \overline{𝔽_{2}}A_{6}-module of dimension 2. Since there is no such module, one concludes that V is not of quadratic type. Now let φ := (V, q) be a 4-dimensional simple orthogonal 𝔽_{3}A_{5}-module. Then there is a linear representation 2.A_{5} → C_{0}(φ)^{∗} giving rise to a nontrivial action of 𝔽̄_{3}2.A_{5} on the 2-dimensional simple C_{0}(φ)-module. Since the two irreducible 𝔽̄_{3}2.A_{5}-modules of dimension 2 are only realisable over 𝔽_{9}, the determinant d_{±}(φ) = −1 is not a square in 𝔽_{3}^{∗}.

5) Let G = U_{3}(5) and φ := (V, q) be a 21-dimensional simple orthogonal ℚG-module. The universal covering group of G is 3.G. Therefore P_{0} : G → c(φ) can be chosen to be linear. There is a unique character χ_{W} of G satisfying χ_{W} ⊗ χ_{W} = χ̃. In the notation of [Conway et al. 1985] one has χ_{W} = 2χ_{1}+χ_{2}+2χ_{3}+2χ_{7}+2χ_{10}+χ_{11}+χ_{12}+χ_{13}+χ_{14}. The character field of χ_{2} (of degree 20) is ℚ and its rational Schur index is 2. If U is the irreducible ℚG-module with character 2χ_{2} then End_{ℚG}(U) = Q_{∞,5} the rational quaternion algebra ramified only at 5 and the infinite place. Now Corollary 3.6 yields [c(φ)] = [Q_{∞,5}].

Let q be a power of an odd prime p. As noted by a referee, the group U_{3}(q) has a absolutely irreducible rational representation V of degree q^{2}−q+1 (cf. [Simpson and Frame 1973]) and a rational character χ of degree q(q−1) with Schur index 2 at ∞ and p ([Gross 1990, Section 14]). So one might hope to generalize this calculation to arbitrary prime powers q. For q = 7 the character χ occurs with odd multiplicity in χ_{W}, so here [c(φ)] = [Q_{∞,7}]. If q = 3 or q = 11, then dim(V) ≡ ^{±}1 (mod 8). Therefore ℝ splits c(φ) for any positive definite quadratic form on V and χ can not occur with odd multiplicity in χ_{W}. But here one finds that the trivial character has odd multiplicity in χ_{W}, hence [c(φ)] = 1 in these cases. For q = 9 the calculations do not allow to determine [c(φ)] uniquely. It seems to be very difficult to calculate the candidates for χ_{W} (for q = 7, 9, 11 one has two possibilities for the character χ_{W}) genericly.

As a referee pointed out, some of the examples above can also be considered from the integral lattice point of view, replacing the determinant by the determinant module L^{#}/L, where L is an G-invariant lattice in the ℚG-module V and L is integral, i.e. contained in its dual lattice L^{#} := {v ∈ V | B_{q}(v, L) ⊆ ℤ}. If L is a maximal integral G-invariant lattice in V, then L^{#}/L is a direct sum of simple selfdual 𝔽_{p}Gmodules, where p runs through the primes dividing |L^{#}/L|. If all selfdual p-modular constituents of V have even degree, then p does not divide the determinant of V. This observation immediately yields that det(φ) = 1 in Example 1), since the character is absolutely irreducible modulo every prime p. Similarly the only primes p that can divide det(φ) are 3 and 5 in Example 2) and 3 in Example 3). In Example 3) one even can conclude that det(φ) = 3, because otherwise G would fix an even unimodular lattice L ⊂ V, and hence 8 | dim(V).

Conclusion. This method is better called a trick, because it can not be applied too often. For instance for the finite simple groups only the first two or three characters χ_{V} usually have a real chance that there is such a constituent ψ of χ_{W} with odd multiplicity. But when this trick can be applied, the calculations are easy and it is surprising to see the determinant of a G-invariant quadratic form appear in the character table.

References

[Conway et al. 1985] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups. Oxford University Press 1985.

[Fröhlich 1972] A. Fröhlich, Orthogonal and symplectic representations of groups. Proc. London Math. Soc. (3) 24 (1972) 470-506.

[GAP] M. Schönert (ed.), Groups, Algorithms and Programming. Lehrstuhl D für Mathematik, RWTH Aachen, Germany (1994) available via anonymous ftp on the server ftp://ftp-gap.dcs.st-and.ac.uk/pub/gap

[Gagola and Garrison 1982] S. M. Gagola and S. C. Garrison, Real characters, double covers and the multiplier. J. Algebra 74 (1982), 20-51.

[Gross 1990] B.H. Gross, Group representations and lattices. J. Amer. Math. Soc. 3 (1990) 929-960.

[Hasse 1924] H. Hasse, Äquivalenz quadratischer Formen in einem beliebigen Zahlkörper. J. für die reine und angew. Math. 153 (1924) 158-162.

[Jansen et al. 1995] C. Jansen, K. Lux, R. A. Parker, R. A. Wilson, An atlas of Brauer Characters. Clarendon Press, Oxford 1995.

[Knus 1991] M.-A. Knus, Quadratic and Hermitian Forms over Rings. Springer Grundlehren 294 (1991)

[Nebe 1999] G. Nebe, Orthogonale Darstellungen endlicher Gruppen und Gruppenringe. Habilitationsschrift RWTH Aachen, Aachener Beiträge zur Mathematik, Band 26, Verlag Mainz (1999)

[Nebe 2000] G. Nebe, Orthogonal Frobenius reciprocity. J. Algebra 225, 250-260 (2000).

[Reiner 1975] I. Reiner, Maximal Orders. Academic Press, 1975.

[Scharlau 1985] W. Scharlau, Quadratic and Hermitian Forms. Springer Grundlehren 270 (1985)

[Simpson and Frame 1973] W. A. Simpson, J. S. Frame, The character tables for SL(3, q), SU(3, q), P SL(3, q), P SU(3.q) Can. J. Math., Vol. XXV, No. 3 (1973) 486-494.

[Turull 1992] A. Turull, The Schur index of projective characters of symmetric and alternating groups. Annals of Math. 135 (1992) 91-124.

Gabriele Nebe, Lehrstuhl B für Mathematik, RWTH Aachen, Templergraben 64, 52062 Aachen, Germany (gabi@math.rwth-aachen.de)