PROJECTIVE INDECOMPOSABLE MODULES, SCOTT MODULES AND THE FROBENIUS-SCHUR INDICATOR

JOHN MURRAY

Abstract. Let Φ be a principal indecomposable character of a finite group G in characteristic 2. The Frobenius-Schur indicator ν(Φ) of Φ is shown to equal the rank of a bilinear form defined on the span of the involutions in G. Moreover, if the principal indecomposable module corresponding to Φ affords a quadratic geometry, then ν(Φ) > 0. This result is used to prove a more precise form of a theorem of Benson and Carlson on the existence of Scott components in the endomorphism ring of an indecomposable G-module, in case the module affords a G-invariant symmetric form.

1. Statement of results

This paper continues our investigation, begun in [7] and [8], into unusual properties of the group algebra of a finite group over a field of characteristic 2. Most of our techniques are not available, and the obvious analogues of our results are false, if the characteristic is odd. The characteristic 2 theory appear to be particularly fertile due to the rich interactions between involutions in the group, the Frobenius-Schur indicator, quadratic forms, and the contragradient operation on the group algebra.

We make extensive use of the modular representation theory of finite groups, as described in [9]. In particular we fix a finite group G and let (O, F, k) be a 2-modular system for G. So O is a complete discrete valuation ring, with field of fractions F, unique maximal ideal J(O) and residue field O/J(O) = k that has characteristic 2. For convenience we assume that both F and k are algebraically closed. We use the symbol R for either of the rings O or k. To avoid trivialities, we assume that |G| is even.

The Frobenius-Schur indicator of a generalized character χ of G is ν(χ) := |G|^{−1} ∑_{g∈G} χ(g^{2}), which turns out to be an integer. If χ is the character of an irreducible F G-module M, then ν(χ) = 1, −1 or 0, depending on whether M is of quadratic, symplectic or not selfdual type, respectively. G. Frobenius and I. Schur first noted this and the fact that |Ω| = ∑_{χ∈Irr(G)} χ(1)ν(χ), where

Ω := {g ∈ G | g^{2} = 1_{G}}.

PROJECTIVE MODULES, SCOTT MODULES AND FROBENIUS-SCHUR INDICATOR 2

Suppose that e is a primitive idempotent in kG. Then there exists a primitive idempotent ê in OG such that e is the image of ê, modulo J(O)G. The module ekG is called a principal indecomposable kG-module, while êOG is called a princpal indecomposable OG-module. The character Φ of F ⊗_{O} P̂ is called the principal indecomposable character of G corresponding to e, ê, ekG or P̂. We may write Φ = ∑ d_{χ,Φ}χ, where χ ranges over the ordinary irreducible characters of G. The non-negative integers d_{χ,Φ} are known as the decomposition numbers of Φ. G. R. Robinson observed in [10] that ν(Φ) = ∑ d_{χ,Φ}ν(χ) is non-negative. This is false when the characteristic is odd.

Date: March 21, 2006.

1991 Mathematics Subject Classification. 20C20.

Key words and phrases. Green correspondence, quadratic form, symmetric bilinear form, Frobenius-Schur indicator, Broué-Robinson form, Scott module.

Each x ∈ RG is an R-linear combination of the elements of G. We use λ(x) to denote the coefficient of 1_{G} in this sum. The map λ : RG → R is called the standard symmetrizing form on RG. The contragradient operator ^{o} is an involutary algebra anti-automorphism of RG that maps each g ∈ G to its inverse. We use RS to denote the span of a subset S of G in RG. Our main result is:

Theorem 1.1. Let e be a primitive idempotent in kG and let Φ be the corresponding principal indecomposable character of G. Then ν(Φ) is the rank of the bilinear form

λ_{e} : kΩ × kΩ → k, where λ_{e}(s, t) := λ(e^{o}set), for all s, t ∈ Ω.

A conjugacy class of G is said to be real if it contains the inverse of each of its elements, and said to be strongly real if each of its elements is inverted by an involution. The R-lattice spanned by a conjugacy class is an RG-permutation module. Theorem 1.1 allows us to add condition (iv) below to the main result of [8]:

Corollary 1.2. Let B be a 2-block of kG. Then the following are equivalent: (i) B is real and has a strongly real defect class;

(ii) ∑_{χ∈Irr(B)} χ(1)ν(χ) ≠ 0_{F};

(iii) kΩ has a composition factor that belongs to B; (iv) λ(e^{o}tes) ≠ 0_{k} for some primitive idempotent e ∈ B and some s, t ∈ Ω.

In conformity with [7] and [8], we call a 2-block that satisfies any one of these equivalent conditions a strongly real 2-block of G.

Our interest in Theorem 1.1 arose as follows. Let K be a field. A KG-module M is said to have a quadratic geometry if there exists a G-invariant K-valued quadratic form Q on M whose polarization b(m_{1}, m_{2}) := Q(m_{1}+m_{2})−Q(m_{1})−Q(m_{2}), ∀m_{1}, m_{2} ∈ M, is non-degenerate. If char(K) is odd, there is a characterization, due separately to W. Willems and J. G. Thompson, of the quadratic type of a principal indecomposable G-module (and its irreducible head) that makes use of the Frobenius-Schur indicator of any one of the irreducible characters of G whose multiplicity in P̂ is odd [12, Proposition 2.2 and Theorem 2.8]. This result does not hold if char(K) = 2. In particular in characteristic 2 there is no known connection between the type of a principal indecomposable module and the type of its irreducible head. Using Theorem 1.1 and the approach adopted by R. Gow and W. Willems in [3], we prove:

Theorem 1.3. Let e be a primitive idempotent in kG and let Φ be the corresponding principal indecomposable character of G. Suppose that ekG has a quadratic geometry. Then ν(Φ) > 0. In particular, e belongs to a strongly real 2-block of G.

A more precise module theoretic form of this result is given in Corollary 6.5.

Example 1.4. Let G be a finite group of Lie type defined over a field of characteristic 2 and let Φ be a principal indecomposable character of G that is real valued.

We claim that ν(Φ) > 0. For, let P be the principal indecomposable kG-module that corresponds to Φ. Then P is of quadratic type, by a result of Gow and Willems [12, 3.9]. Our claim then follows from Theorem 1.3.

Example 1.5. Let G = H ≀C_{2}, where H is the unique nonabelian group of order 12 that is not isomorphic to A_{4} or a dihedral group. Then [3, 2.12] shows that kG has a principal indecomposable module that does not have a quadratic geometry. However the character Φ of this module satisfies ν(Φ) = 2. So the converse of Theorem 1.3 is false.

Theorem 7.2 is a refinement, for modules that possess a G-invariant symmetric bilinear form, of a result of D. Benson and J. Carlson on the existence of Scott components in the endomorphism ring of a kG-module.

Theorem A.5 is concerned with bilinear forms and projective (in the sense of Schur) modules. This result is needed to prove 7.2. Since Theorem A.5 has a different character to the rest of the paper, we consign its proof to the appendix.

2. Bilinear forms and adjoints

Just as in [7] and [8] we let Σ be a cyclic group of order 2, generated by an involution σ. The wreath product G ≀ Σ of G with Σ is a split extension of the base group G×G by Σ. Here σ acts on G×G via (g_{1}, g_{2})^{σ} = (g_{2}, g_{1}), for all g_{1}, g_{2} ∈ G. If H is a subgroup of G then the diagonal subgroup of G ≀ Σ is \underline{H} := \{(h, h) | h ∈ H\}.

Throughout the paper M will be a right RG-module: the image of m ∈ M under g ∈ G is written m · g. We write endomorphisms and linear forms on the right, but most other functions on the left. We use M↓_{H} for the restriction of M to H, and N↑^{G} for the induced RG-module N ⊗_{RH} RG, whenever N is an RH-module. A theorem of J. A. Green ([4]) states that if M is an indecomposable OG-module, with F -character χ then

(1) χ(g) = 0, if the 2-part of g ∈ G is not contained in some vertex of M.

Let μ : RG → End_{R}(M) be the ring homomorphism associated with M. Then the dual space M^{∗} = Hom(M, R) is an RG-module via f · g := μ(g^{−1})f, for all f ∈ M^{∗} and g ∈ G. Also End_{R}(M) is an RG × G-module, via f · (g_{1}, g_{2}) := μ(g_{1}^{−1})fμ(g_{2}), for all f ∈ End_{R}(M) and g_{1}, g_{2} ∈ G. In particular for the restricted module End_{k}(M)↓\underline{G}, the action of \underline{g} ∈ \underline{G} is conjugation f ·(g, g) := μ(g^{−1})fμ(g) by the unit μ(g) ∈ End_{k}(M). We identify RG × G-modules M^{∗} ⊗_{R} M = End_{R}(M). The space Bil_{R}(M) of all bilinear forms on M is an RG × G-module via (b · (g_{1}, g_{2}))(m_{1}, m_{2}) := b(m_{1} · g_{1}^{−1}, m_{2} · g_{2}^{−1}), for all bilinear forms b, and all m_{1}, m_{2} ∈ M. We identify RG × G-modules M^{∗} ⊗_{R} M^{∗} = Hom(M, M^{∗}) = Bil_{R}(M). Note also the natural isomorphism M ⊗_{R} M ≅ Bil_{R}(M)^{∗}.

The equality Bil_{R}(M) = Hom(M, M^{∗}), identifies a bilinear form b with the map M → M^{∗} that sends m_{2} ∈ M to the linear form m_{1} → b(m_{1}, m_{2}), for all m_{1} ∈ M. We say that b is non-degenerate if this map is an R-isomorphism, and we say that b is G-invariant if this map is an RG-homomorphism. Now M is said to be self-dual if M ≅ M^{∗} as RG-modules. So M is self-dual if and only there exists a non-degenerate G-invariant bilinear form on M. For example, the form B_{1}(x, y) := λ(xy^{o}), on the regular RG-module, is non-degenerate and G-invariant. So RG is a self-dual RG-module.

Let N be an RG-module and let f ∈ Hom(M, N^{∗}). Then f^{t} ∈ Hom(N, M^{∗}) is defined by m(nf^{t}) := n(mf), for all m ∈ M and n ∈ N. If N = M^{∗}, then Hom(M, N^{∗}) = End_{R}(M). In this case we call f^{t} ∈ End_{R}(M) the transpose of f. In terms of tensors, (α ⊗ β)^{t} = β ⊗ α, for all α, β ∈ M^{∗}.

We extend Bil_{R}(M) to a G≀Σ-module by defining b·σ := b^{t}, for each b ∈ Bil_{R}(M). Thus b · σ(m_{1}, m_{2}) := b(m_{2}, m_{1}), for all m_{1}, m_{2} ∈ M. B. Külshammer uses the notation M^{⊗2} for the extension of M⊗_{R}M to G≀Σ, such that m_{1}⊗m_{2}·σ := m_{2}⊗m_{1}, for all m_{1}, m_{2} ∈ M. Clearly Bil_{R}(M) ≅ (M^{∗})^{⊗2}, as RG ≀ Σ-modules. It is shown in [6] that M^{⊗2} is indecomposable if M is indecomposable. Moreover, if M is indecomposable with vertex V, then M^{⊗2} has vertex V ≀ Σ. If R = O and F ⊗_{O} M has character χ, then M^{⊗2} has character χ^{⊗2}, where

χ^{⊗2} ((g_{1}, g_{2})σ) := χ(g_{1}g_{2}), for all g_{1}, g_{2} ∈ G.

Let b be a non-degenerate bilinear form on M and let f ∈ End_{R}(M). Then there is a unique endomorphism f^{β} of M such that b(m_{1}f^{β}, m_{2}) = b(m_{1}, m_{2}f), for all m_{1}, m_{2} ∈ M. We call f^{β} the adjoint of f with respect to b. Clearly the adjoint map f → f^{β} is an R-algebra anti-automorphism of End_{R}(M). Our next lemma shows that a non-degenerate G-invariant bilinear form can be recovered from its adjoint.

Lemma 2.1. The map sending a non-degenerate form b to its adjoint β establishes a bijection between the rank 1-subspaces of Bil_{R}(M) that contain a non-degenerate G-invariant form and the algebra anti-automorphisms of End_{R}(M) that invert each μ(g), with g ∈ G. If R = k then b is symmetric if and only if β is an involution.

Proof. Let b be a non-degenerate G-invariant bilinear form on M, with adjoint map β. The G-invariance of b implies that μ(g)^{β} = μ(g^{−1}), for all g ∈ G. Note that if λ ∈ R, then λb is non-degenerate if and only if λ is a unit in R. Also if λb is non-degenerate then it has adjoint β.

Conversely let γ be an R-algebra anti-automorphism of End_{R}(M) such that μ(g)^{γ} = μ(g^{−1}), for each g ∈ G. Choose a primitive idempotent ϵ in End_{R}(M). Then ϵ^{γ} is also a primitive idempotent in End_{R}(M). Choose an R-isomorphism ϕ : ϵEnd_{R}(M)ϵ^{γ} → R. Now ϵEnd_{R}(M) is an irreducible End_{R}(M)-module that is isomorphic to M as G-module. Define an R-bilinear form c on ϵEnd_{R}(M) by setting c(ϵf_{1}, ϵf_{2}) = ϕ(ϵf_{1}f_{2}^{γ}ϵ^{γ}), for all f_{1}, f_{2} ∈ End_{R}(M). Then c is non-degenerate, as its kernel is a proper End_{R}(M)-submodule of ϵEnd_{R}(M). Clearly c has adjoint map γ. In addition, c is G-invariant, as μ(g)^{γ} = μ(g^{−1}), for each g ∈ G.

Let b and c be non-degenerate G-invariant bilinear forms on M, whose adjoints coincide with β. Let B : M → M^{∗}, C : M → M^{∗} be the G-module isomorphisms corresponding to b, respectively c. Then f^{β} = Bf^{∗}B^{−1} and also f^{β} = Cf^{∗}C^{−1}, for all f ∈ End_{R}(M). So B^{−1}fB = C^{−1}fC, for all f ∈ End_{R}(M). It follows that CB^{−1} is a central unit in End_{R}(M), whence C = λB, for some unit λ in R. This shows that the correspondence Rb ↔ β is bijective.

If b is symmetric then β is easily seen to be an involution. Suppose that R = k and that β is an involution. Then β acts as an involutary anti-automorphism on the 1-dimensional k-space ϵEnd_{k}(M)ϵ^{β}. As char(k) = 2, this map must be the identity. We conclude from this that b is symmetric. □

Proposition 2.2. Suppose that M affords a nondegenerate G-invariant symmetric bilinear form b. Let β be the adjoint of b. Then End_{R}(M) can be extended to a G ≀ Σ-module by letting σ act as β on End_{R}(M). Moreover End_{R}(M) ≅ Bil_{R}(M), as RG ≀ Σ-modules.

Proof. It is easily checked that f · σ := f^{β}, for all f ∈ End(M), extends the G × Gaction to G ≀ Σ. The required G ≀ Σ-module isomorphism sends f ∈ End_{R}(M) to B_{f} ∈ Bil_{R}(M), where B_{f}(m_{1}, m_{2}) := b(m_{1}f, m_{2}), for all m_{1}, m_{2} ∈ M. □

3. A Scott multiplicity formula

Let H be a subgroup of G. We use M^{H} to denote the space of H-fixed points in M, but we also use the alternatives Bil_{RH}(M) and End_{RH}(M). The relative trace map Tr_{H}^{G} : M^{H} → M^{G} is defined by Tr_{H}^{G}(m) := ∑ m · g, for all m ∈ M^{H}. Here g ranges over a set of representatives for the right cosets of H in G. Set Tr_{H}^{G}(M^{H}) := \{Tr_{H}^{G}(m) | m ∈ M^{H}\}. We shall identify the groups \underline{H} × Σ and H × Σ in expressions involving the relative trace map on \underline{G} × Σ-modules. For instance Tr_{<gσ>}^{G×Σ} is the trace map from < (g, g)σ > to \underline{G} × Σ.

The Scott module S_{G}(H) is the only component of R_{H}↑^{G} that has a trivial submodule or a trivial factor module (c.f. [9, 4.8.4]). It is known that each Sylow 2-subgroup of H is a vertex of S_{G}(H). J. A. Green proved the following in [5, (1.3)]:

Lemma 3.1. The multiplicity of the Scott module with vertex V ≤ G as a component of M is the rank of the bilinear form ρ_{V,M} : Tr_{V}^{G}(M^{V}) × Tr_{V}^{G}((M^{∗})^{V}) → k, where

ρ_{V,M}(m, f) = mf_{1} = m_{1}f,

whenever m = Tr_{V}^{G}(m_{1}) for m_{1} ∈ M^{V}, and f = Tr_{V}^{G}(f_{1}) for f_{1} ∈ (M^{∗})^{V}.

Remark 3.2. The form ρ_{V,M} is well-behaved with respect to direct products. Specifically, suppose that M = M_{1} ⊕ M_{2} as kG-modules, that m_{1} ∈ Tr_{V}^{G}(M^{V}) ∩ M_{1}, and that ρ_{V,M}(m_{1}, f) ≠ 0_{k}, where f ∈ Tr_{V}^{G}((M^{∗})^{V}). Write f = f_{1} + f_{2}, where f_{i} is the projection of f onto M_{i}^{∗}. Then ρ_{V,M}(m_{1}, f) = ρ_{V,M}(m_{1}, f_{1}) ≠ 0_{k}. In particular, in this situation M_{1} has a Scott component with vertex V.

The next result is a consequence of Mackey’s formula.

Lemma 3.3. Suppose that V and W are 2-subgroups of G such that no G-conjugate of W contains V. Then

m_{1}f = 0_{k}, if m_{1} ∈ M^{V} and f ∈ Tr_{W}^{G} ((M^{∗})^{W});
mf_{1} = 0_{k}, if m ∈ Tr_{V}^{G}(M^{V}) and f_{1} ∈ (M^{∗})^{W}.

We note also that:

Lemma 3.4. Suppose that no component of M has a vertex that properly contains V ≤ G. Then ρ_{V,M} extends to a bilinear form ρ̂_{V,M} on M^{G} × (M^{∗})^{V}, such that ρ̂_{V,M}(m, Tr_{V}^{G}(f_{1})) = mf_{1}, for all m ∈ M^{G} and f_{1} ∈ (M^{∗})^{V}. The rank of ρ̂_{V,M} equals the rank of ρ_{V,M}.

Now suppose that A is a symmetric G-algebra, with symmetrizing form t, and let D be a 2-subgroup of G. M. Broué and G. R. Robinson [2, (1.2)] define the symmetric bilinear form ρ_{D} = ρ_{D,G}^{A,t} on Tr_{D}^{G}(A^{D}) as

ρ_{D}(x, y) = t(x_{1}y) = t(xy_{1}),

whenever x = Tr_{D}^{G}(x_{1}), or y = Tr_{D}^{G}(y_{1}), with x_{1}, y_{1} ∈ A^{D}. Using Green’s result, Lemma 3.1, they show that the rank of ρ_{D} coincides with the multiplicity of the Scott module with vertex D as a component of A.

Now take A = End_{k}(M) and regard End_{k}(M) as a G-algebra via the restriction of the G × G-module End_{k}(M) to \underline{G}. Let t = tr denote the usual trace form on End_{k}(M). Set ρ_{V} := ρ_{V,G}^{End_{k}(M),tr}.

Proposition 3.5. Suppose that M affords a nondegenerate G-invariant symmetric bilinear form b, and that End_{k}(M) is extended to a G ≀ Σ-module, according to Proposition 2.2. Let D̂ be a 2-subgroup of \underline{G} × Σ. Set \underline{D} = D̂ ∩ \underline{G}. Then the multiplicity of the Scott module with vertex D̂ as a component of End_{k}(M)↓_{G×Σ} is equal to the rank of the restriction ρD̂ of ρ_{D} to TrG×Σ ; D̂(End_{kD̂}(M)).

Proof. We may assume that D̂ ≠ \underline{D}. Note that the restriction makes sense. For, D̂ = \underline{D} < \underline{t}σ >, where t is any element of D̂\\underline{D}. Any set of representatives for the cosets of \underline{D} in \underline{G} is also a set of representatives for the cosets of \underline{D} < \underline{t}σ > in \underline{G} × Σ.

We adapt the proof of Proposition (1.3) in [2]. Let {m_{i}} be a basis of M, with b-dual basis {n_{i}}. So b(m_{i}, n_{j}) = δ_{ij}, for all i and j. As b is symmetric, {m_{i}} is the b-dual basis of {n_{i}}. Now for f ∈ End_{k}(M) we have tr(f) = ∑_{i} b(m_{i}f, n_{i}). Thus

tr(f^{β}) = ∑_{i} b(m_{i}f^{β}, n_{i}) = ∑_{i} b(m_{i}, n_{i}f) = ∑_{i} b(n_{i}f, m_{i}) = tr(f).

For f_{2} ∈ End_{k}(M), define f_{2}T ∈ End_{k}(M)^{∗} by f_{1}(f_{2}T) = tr(f_{1}f_{2}), for all f_{1} ∈ End_{k}(M). Then T is a \underline{G}-module isomorphism End_{k}(M) → End_{k}(M)^{∗}. Also

f_{1}((f_{2}T)^{β}) = f_{1}^{β}(f_{2}T) = tr(f_{1}^{β}f_{2}) = tr(f_{2}^{β}f_{1}) = tr(f_{1}f_{2}^{β}) = f_{1}(f_{2}^{β}T),

for all f_{1} ∈ End_{k}(M). So (f_{2}T)^{β} = f_{2}^{β}T and hence T is even a \underline{G} × Σ-module isomorphism. In particular, if H ≤ \underline{G} × Σ then the image of End_{kH}(M) under T is End_{kH}(M)^{∗}.

By Lemma 3.1 the multiplicity of the Scott module with vertex D̂ as a component of End_{k}(M)↓_{G×Σ} is the rank of the bilinear form ρD̂ on TrG×Σ ; D̂(End_{kD̂}(M)), where ρD̂(x, y) = tr(x_{1}y) = tr(xy_{1}), whenever x = TrG×Σ ; D̂(x_{1}), or y = TrG×Σ ; D̂(y_{1}), with x_{1}, y_{1} ∈ End_{kD̂}(M). The lemma now follows from the observation that ρD̂ coincides with the restriction of ρ_{D} to TrG×Σ ; D̂(End_{kD̂}(M)). □

4. Bilinear forms on the group algebra

Recall that λ : RG → R, with λ(∑ μ_{g}g) = μ_{1}, is a symmetrizing form on RG. The corresponding bilinear form B_{1}(x, y) := λ(xy^{o}) is G-invariant, symmetric and non-degenerate. So End_{RG}(RG) ≅ Bil_{R}(RG), as G ≀ Σ-modules. Concretely, x · (g_{1}, g_{2}) := g_{1}^{−1}xg_{2} and x · σ := x^{o}, for each x ∈ RG and g_{1}, g_{2} ∈ G. We use the isomorphism RG ⊗_{R} RG ≅ Bil_{R}(RG), without further comment.

Lemma 4.1. Each non-projective component of Bil_{R}(RG) has vertex Σ and takes the form P^{⊗2}, for some principal indecomposable RG-module P; the multiplicity of P^{⊗2} equals the dimension of the corresponding irreducible kG-module.

Proof. Let 1_{G} = e_{1} + … + e_{d} + … + e_{m} be a decomposition of 1_{G} into a sum of pairwise orthogonal primitive idempotents in RG. Then

RG ⊗_{R} RG = ∑_{i}(e_{i}RG)^{⊗2} + ∑_{i<j}(e_{i}RG ⊗ e_{j}RG + e_{j}RG ⊗ e_{i}RG).

Each term in the second sum is a projective G ≀ Σ-module. The lemma follows from this. □

Lemma 4.2. Bil_{R}(RG) ≅ R_{Σ}↑^{G≀Σ}.

Proof. Clearly {g_{1}⊗g_{2} | g_{1}, g_{2} ∈ G} is a G≀Σ-permutation basis for RG^{⊗2}. Moreover G ≀ Σ acts transitively on this basis and the stabilizer of 1_{G} ⊗ 1_{G} is Σ. □

For g ∈ G, define g^{∗} ∈ (RG)^{∗} by gg^{∗} = 1_{R} and hg^{∗} = 0_{R}, for g ≠ h ∈ G. Then {g_{1}^{∗} ⊗ g_{2}^{∗} | g_{1}, g_{2} ∈ G} forms a basis for Bil_{R}(RG). Now for x ∈ G, we have g^{∗}·x = (gx)^{∗}, in the dual G-module (RG)^{∗}. From this it follows that Tr_{1}^{G}(g_{1}^{∗}⊗g_{2}^{∗}) = B_{g_{1}g_{2}^{−1}}, where B_{a}(x, y) := λ(axy^{o}), for all a, x, y ∈ RG. Thus {B_{g} | g ∈ G} is a basis for the space Bil_{RG}(RG) of G-invariant bilinear forms on RG. Clearly B_{a} is a symmetric form if and only if a = a^{o}. Let (G\Ω)^{±} be a set of representatives for the subsets {g, g^{−1}}_{g∈G} of G\Ω. Then {B_{t} | t ∈ Ω} ∪ {B_{g+g^{−1}} | g ∈ (G\Ω)^{±}} is a basis for the space Bil_{RG×Σ}(RG) of G-invariant symmetric bilinear forms on RG. Also if e is an idempotent in RG, then

(2)  Bil_{RG}(eRG) = {B_{e^{o}ae} | e^{o}ae ∈ e^{o}RGe}.

Now let R = k and choose t ∈ Ω. Let T be the conjugacy class of G that contains t. Recall that Bil_{k}(kG)^{∗} ≅ kG ⊗_{k} kG. For < \underline{t}σ >-fixed points

(3) (kG ⊗_{k} kG)^{k<tσ>} has k-basis {gt ⊗ g | g ∈ G} ∪ {g_{1}t ⊗ g_{2} + g_{2}t ⊗ g_{1} | g_{1} ≠ g_{2} ∈ G}.

The analogous basis of Bil_{k<tσ>}(kG) enables one to show that

(4) Tr_{<tσ>}^{G×Σ} (Bil_{k<tσ>}(kG)) has k-basis {B_{s} | s ∈ T } ∪ {B_{g+g^{−1}} | g ∈ (G\Ω)^{±}}.

Lemma 4.3. Let e be a primitive idempotent in kG, let t ∈ Ω and let T be the conjugacy class of G that contains t. Then the multiplicity of the Scott module with vertex < \underline{t} σ > as a component of Bil_{k}(ekG)↓_{\underline{G}×Σ} coincides with the rank of the symmetric bilinear form

λ_{e,T} : kT × kT → k, where λ_{e,T} (r, s) := λ(e^{o}res), for all r, s ∈ T .

Proof. Let s ∈ T and g ∈ G. Then B_{e^{o}se}(egt ⊗ eg) = λ(e^{o}segtg^{−1}). The result now follows from (2), (3) and (4), and Lemmas 3.1 and 3.3. □

Let T_{0} = {1_{G}} and let T_{1} …, T_{n} be the conjugacy classes of involutions in G. We extend each λ_{e,T_{i}} to a symmetric bilinear form on kΩ by setting λ_{e,T_{i}}(s, t) = 0_{k}, whenever s ∉ T_{i} or t ∉ T_{i}. Recall the definition of λ_{e}, from the statement of Theorem 1.1.

Proposition 4.4. Let e be a primitive idempotent in kG. Then

λ_{e} = λ_{e,T_{0}} ⊥ λ_{e,T_{1}} ⊥ … ⊥ λ_{e,T_{n}}.

The rank of λ_{e} is the number of nonprojective Scott components in Bil_{k}(kG)↓_{\underline{G}×Σ}.

Proof. Suppose that s, t ∈ Ω are not conjugate in G. Then < \underline{t}σ > is not contained in any \underline{G} × Σ-conjugate of < \underline{s}σ >. So λ(e^{o}tes) = B_{e^{o}te}(es ⊗ e) = 0_{k}, using Lemma 3.3. The result now follows from Lemmas 4.1 and 4.3. □

5. Scott multiplicities from the Frobenius-Schur indicator

In this section we aim to interpret a result of G. R. Robinson on principal indecomposable modules in characteristic 2. We give a Scott-multiplicity formula in terms of the restriction of a projective character to the centralizer of an involution. We then relate this to the Frobenius-Schur indicator of the projective character.

Let t ∈ Ω and set T :=< \underline{t}σ >. Then C_{G}(t) ≅ (\underline{C_{G}(t)} × Σ)/T. Suppose that Q is a principal indecomposable RC_{G}(t)-module. Denote by Q̂ the inflation of Q, regarded as an \underline{C_{G}(t)}×Σ/T -module, to \underline{C_{G}(t)}×Σ. Then Q̂ is indecomposable with vertex T and its kernel contains T. Conversely, each indecomposable \underline{C_{G}(t)} × Σmodule that has vertex T and kernel containing T has the form P̂, for some principal indecomposable RC_{G}(t)-module P. We use fQ to denote the Green correspondent, with respect to (\underline{G} × Σ, T, \underline{C_{G}(t)} × Σ), of Q̂. So fQ is an R\underline{G} × Σ-module that has trivial source and vertex T. Moreover, fQ is the unique non-projective component of Q̂↑^{\underline{G}×Σ}, and Q̂ is the unique component of fQ↓_{\underline{C_{G}(t)}×Σ} that has vertex T. Note that for each involution s ∈ \underline{C_{G}(t)} that is G-conjugate to t, the restricted module fQ↓_{\underline{C_{G}(t)}×Σ} has at least one component with vertex < \underline{s}σ >.

Given g ∈ G we may write g = g_{2}g_{2′} = g_{2′}g_{2}, for a unique 2-element g_{2} and a unique 2′-element g_{2′} in G. The Frobenius twist M^{Fr} of M is the RG-module with the same underlying R-module M, where g ∈ G acts on M^{Fr} as g_{2}g_{2′}^{2} acts on M. If M has (Brauer or ordinary) character ϕ then M^{Fr} has character ϕ^{Fr} : g → ϕ(g_{2}g_{2′}^{2}), for all g in the domain of ϕ.

We use Φ_{Q} to denote the character of F ⊗_{O} Q whenever H is a subgroup of G and Q is an OH-module. Our next result is more general than required here.

Lemma 5.1. Let P be a principal indecomposable OG-module, let t ∈ Ω, and let {Q} range over the isomorphism classes of principal indecomposable OC_{G}(t)- modules. Then

P^{Fr}↓_{C_{G}(t)} = ∑ a_{Q}Q,  if and only if  P^{⊗2}↓_{\underline{G}×Σ} = ∑ a_{Q}fQ.

Proof. Suppose that Q is a principal indecomposable OC_{G}(t)-module. Then Q̂ is the unique component of fQ↓_{\underline{C_{G}(t)}×Σ} that has a vertex containing < \underline{t}σ >, and

< \underline{t}σ > is contained in the kernel of Q̂. It then follows from (1) that Φ_{fQ}(\underline{gt}σ) = ΦQ̂(\underline{gt}σ) = Φ_{Q}(g), for each 2′-element g ∈ C_{G}(t). Thus

(Φ^{⊗2}−∑a_{Q}Φ_{fQ}) (\underline{gt}σ) = Φ(g^{2})−∑a_{Q}Φ_{Q}(g) = (Φ^{Fr}↓_{C_{G}(t)}−∑a_{Q}Φ_{Q}) (g) = 0,

for each 2-regular element g in C_{G}(t).

The functions Φ_{Q} are linearly independent on the 2′-elements of C_{G}(t). It follows that the functions Φ_{fQ} are linearly independent on the 2-section of \underline{G} × Σ that contains \underline{t}σ. Moreover, if an indecomposabe O\underline{G} × Σ-module has a character that does not vanishes on the 2-section of \underline{G}×Σ that contains \underline{t}σ then by (1) that module has a vertex that contains \underline{t}σ. The proposition now follows from the previous paragraph. □

Corollary 5.2. Let P be a principal indecomposable OG-module and let Φ be the character of F ⊗_{O} P. Then for t ∈ Ω, the Scott module with vertex < \underline{t} σ > occurs with multiplicity < Φ↓_{C_{G}(t)}, 1_{C_{G}(t)} > as a component of P^{⊗2}↓_{\underline{G}×Σ}.

Proof. Let Q be the projective cover of the trivial OC_{G}(t)-module. Then Q is the Scott module with trivial vertex for C_{G}(t) and Q̂ is the Scott module with vertex < \underline{t}σ > for \underline{C_{G}(t)}×Σ. Green correspondence preserves Scott modules. So fQ is the Scott module with vertex < \underline{t}σ > for \underline{G}×Σ. The trival F C_{G}(t)-module occurs with multiplicity 1 as a submodule of F ⊗_{O} Q, and with multiplicity 0 as a submodule of F ⊗_{O} Q′, for any principal indecomposable OC_{G}(t)-module Q′ ≅ Q̸. It follows that Q occurs with multiplicity < Φ^{Fr}↓_{C_{G}(t)}, 1_{C_{G}(t)} >=< Φ↓_{C_{G}(t)}, 1_{C_{G}(t)} > as a component of P^{Fr}↓_{C_{G}(t)}. The result now follows from Lemma 5.1. □

Proof of Theorem 1.1. Recall that Ω is a union of the G-conjugacy classes ⋃_{i=0}^{n} T_{i}. Choose t_{i} ∈ T_{i}, for i = 0, …, n. It follows from Corollary 5.2 and Proposition 4.4 that λ_{e} has rank equal to ∑_{i=0}^{n} < Φ↓_{C_{G}(t_{i})}, 1_{C_{G}(t_{i})} >. The proof is now a consequence of G. R. Robinson’s observation [10, Lemma 1] that ν(Φ) = ∑_{i=0}^{n} < Φ↓_{C_{G}(t_{i})}, 1_{C_{G}(t_{i})} >. □

6. Quadratic forms and the Frobenius-Schur indicator

In this section we adopt the approach of Gow and Willems to quadratic forms on principal indecomposable RG-modules in order to prove Theorem 1.3. We highlight two results from [3] that will be important for our purposes.

Lemma 6.1. No principal indecomposable OG-module has a symplectic geometry.

Proof. Suppose for the sake of contradiction that P is a principal indecomposable OG-module that has a nondegenerate G-invariant symplectic bilinear form b. Then b induces a symplectic form, also denoted by b, on F ⊗_{O} P. Proposition 1.1 of [3] implies that there is an irreducible F G-module M such that M is of quadratic type and M occurs with odd multiplicity in F ⊗_{O} P. Then Lemma 3.6 of [12] shows that there is a component M′ of F ⊗_{O} P, that is isomorphic to M, such that the restriction of b to M′ is nondegenerate. Thus M is of quadratic type and also of symplectic type, a contradiction. □

Lemma 6.2. Let P be a principal indecomposable kG-module. Then each nondegenerate G-invariant quadratic form on P can be extended to a non-degenerate G-invariant quadratic form on kG. If in addition P is not the projective cover of k_{G}, then each G-invariant symmetric form on P is the polarization of a G-invariant quadratic form on P.

Proof. This follows from Propositions 2.2 and 2.6 in [3]. □

We say that a ∈ RG is symmetric if a = a^{o}, and say that it is even if λ(a) ∈ 2R. When dealing with quadratic forms on RG it is useful to fix (arbitrarily) a total order < on the elements of G. Suppose that a = ∑_{g∈G} a_{g}g ∈ RG is even and symmetric. Then for each s ∈ R, define a quadratic form Q_{s,a} on RG via

Q_{s,a}( ∑_{g∈G} x_{g}g) := s ∑_{g∈G} x_{g}^{2} + ∑_{h<i∈G} x_{h}x_{i}a_{ih^{−1}}. This is well defined because a = a^{o}.

Moreover it is known that

{Q_{s,a} | s ∈ R, and a ∈ RG, even and symmetric}

gives all G-invariant quadratic forms on RG. If R = k then

(5)  B_{a}(x, y) = Q_{s,a}(x + y) − Q_{s,a}(x) − Q_{s,a}(y), for all x, y ∈ kG,

is the polarisation of Q_{s,a}.

Corollary 6.3. Let e be a primitive idempotent in kG. Then ekG has a quadratic geometry if and only if there exists a ∈ kG, even and symmetric, such that the restriction of B_{a} to ekG is nondegenerate.

Proof. Suppose first that ekG is the projective cover of the trivial module. Then ekG has multiplicity 1 as a component of kG. It follows from this that if t ∈ Ω then the restriction of B_{t} is a nondegenerate G-invariant symmetric bilinear form on ekG.

Now suppose that ekG is not the projective cover of the trivial module. Then the desired conclusion follows from Lemma 6.2 and the above description of the G-invariant quadratic forms on kG. □

The proof of the following result is adapted from that of Lemma 3.2 in [3]:

Lemma 6.4. Let e be a primitive idempotent in kG. Suppose that a ∈ kG is even and symmetric and that the restriction of B_{a} to ekG is nondegenerate. Then there exists t ∈ Ω such that λ(at) ≠ 0_{k}, and the restriction of B_{t} to ekG is nondegenerate.

Proof. As Soc(ekG) is irreducible, the degeneracy of a bilinear form on ekG depends on whether or not Soc(ekG) is contained in its kernel. It follows that if a = c + d where c, d ∈ kG, then the restriction of one of B_{c} or B_{d} to ekG is nondegenerate.

Write a = c + d where c = ∑_{t∈Ω\{1}} λ(at)t and d = ∑_{g∈(G\Ω)^{±}} λ(ag)(g + g^{−1}).

We claim that B_{d} is degenerate. Suppose otherwise. Set d̂:= ∑_{g∈(G\Ω)^{±}} { \atop λ(ag)}(g −

g^{−1}) ∈ OG, where { \atop λ(ag)} ∈ O has image λ(ag) modulo J(O). Then d̂^{o} = −d̂ and d is the image of d̂ modulo J(O)G. As d̂ is skew-symmetric, Bd̂ is a non-degenerate G-invariant symplectic form on the lift êkG of ekG to OG. This contradicts Lemma 6.1, and proves our claim. It now follows from the first paragraph that there exists t ∈ Ω such that λ(at) ≠ 0_{k} and the restriction of B_{t} to ekG is nondegenerate. □

Proof of Theorem 1.3. Corollary 6.3 implies that there exists a ∈ kG such that a is even and symmetric and the restriction of B_{a} to ekG is nondegenerate. It then follows from Lemma 6.4 that there exists t ∈ Ω such that the restriction of B_{t} to ekG is nondegenerate. Now the restriction of B_{t} to ekG coincides with the restriction of B_{e^{o}te} to ekG. So, again using Lemma 6.4, there exists s ∈ Ω such that λ((e^{o}te)s) ≠ 0_{k}. We conclude from Theorem 1.1 that ν(Φ) > 0. □

Theorem 3.1 of [3] states that a principal indecomposable RG-module P has a quadratic geometry if and only if there exists a primitive idempotent e ∈ RG, and an element t ∈ Ω, such that P ≅ eRG and e^{o} = tet. We note the following consequence of our methods:

Corollary 6.5. Let e be a primitive idempotent in kG and let t ∈ Ω be such that e^{o} = tet. Then the irreducible kG-module ekG/J(ekG) occurs as a composition factor in k_{C_{G}(t)}↑^{G}.

Proof. The essential work in the proof of Lemma 3.5 of [3] is to show that if e^{o} = tet, then the restriction of B_{t} to ekG is nondegenerate. As above, this means that there exists s ∈ Ω such that λ(e^{o}tes) ≠ 0_{k}. Proposition 4.4 forces s to be G-conjugate to t. We deduce from this and Lemma 4.3 that the Scott module with vertex < \underline{t} σ > is a component of ekG^{⊗2}↓_{\underline{G}×Σ}. It then follows from Corollary 5.2 that the projective cover of the trivial kC_{G}(t)-module is a component of the restriction ekG↓_{C_{G}(t)}. Then by Frobenius-Nakayama reciprocity [9, 3.1.27(i)] the irreducible module ekG/J(ekG) is a composition factor of k_{C_{G}(t)}↑^{G}. □

7. Extension of a theorem of Benson and Carlson

In this section M is an indecomposable kG-module that affords a non-degenerate G-invariant symmetric bilinear form b. The adjoint β of b is an involutary kalgebra anti-automorphism of End_{k}(M), such that μ(g)^{β} = μ(g^{−1}), for all g ∈ G. Proposition 2.2 implies that End_{k}(M) ≅ M^{⊗2}, as kG ≀ Σ-modules. Here f · σ = f^{β}, for all f ∈ End_{k}(M). Using the methods of Section 6, we prove an analogue of a theorem of Benson and Carlson on the existence of Scott components in End_{k}(M).

Fix a vertex V of M, and a V -source S of M. Then V × V is a vertex of End_{k}(M), as G×G-module. By Mackey’s formula, each component of End_{k}(M)↓\underline{G} has a vertex contained in \underline{V}. D. Benson and J. Carlson prove in [1, 2.4] that (6) End_{k}(M)↓\underline{G} has a Scott component with vertex \underline{V} if and only if dim(S) is odd.

Now V ≀ Σ is a vertex of End_{k}(M), as G ≀ Σ-module. Again by Mackey’s formula, each component of End_{k}(M)↓_{\underline{G}×Σ}, has a vertex contained in a group of the form \underline{V} < \underline{n}σ >, where n ∈ N_{G}(V) is such that n^{2} ∈ V. In view of (6), we ask

Question 7.1. Does End_{k}(M)↓_{\underline{G}×Σ} have a Scott component with vertex \underline{V} < \underline{n}σ > for some n ∈ N_{G}(V) with n^{2} ∈ V?

If the answer is ‘yes’, then in particular End_{k}(M)↓\underline{G} has a Scott component with vertex \underline{V}. So dim(S) is odd. We therefore assume from now on that dim(S) is odd.

Proposition 3.5 shows that Question 7.1 can be answered by studying the restriction of the Broué-Robinson form to a certain subspace of Tr_{V}^{G}(End_{kV} (M)).

L. Puig defines a point of an algebra A to be an A^{×}-conjugacy class of primitive idempotents of A. The theory of points and the related notions of defect points, multiplicity modules and multiplicity algebras is comprehensively explained in [11]. We borrow heavily from Thevenáz book.

Let δ_{1} be the defect point of the G-algebra End_{k}(M) corresponding to the V - source S of M. So Me ≅ S as V -modules, for any idempotent e ∈ δ_{1}. The inertial group of S or of δ_{1} in N_{G}(V)/V is I := {g ∈ N_{G}(V) | S^{g} ≅ S}/V. Let 𝔐_{1} be the unique maximal ideal of End_{kV} (M) that does not contain any idempotent in δ_{1}. The simple quotient algebra End_{kV} (M)/𝔐_{1} is called a defect multiplicity algebra of End_{k}(M). By Wedderburn’s theorem, this algebra is the endomorphism algebra of a defect multiplicity module P_{1} of End_{k}(M). It is known that P_{1} is a projective indecomposable module for a twisted group algebra of I.

Now σ acts on End_{kV} (M). Set δ_{2} := {e^{σ} | e ∈ δ_{1}}. Then δ_{2} is a defect point of End_{kV} (M) and Me ≅ S^{∗} as V -modules, for each idempotent e ∈ δ_{2}. Let P_{2} be the defect multiplicity module of End_{k}(M) corresponding to δ_{2}. Its endomorphism ring is End_{kV} (M)/𝔐_{2}, where 𝔐_{2} := 𝔐_{1}^{σ}.

Define the extended inertial group of S or of δ_{1} in N_{G}(V)/V as

J := {g ∈ N_{G}(V) | S^{g} ≅ S or S^{g} ≅ S^{∗}}/V.

Note that I ≤ J and that [J : I] = 1 or 2. For the moment we assume that [J : I] = 2.

Set P := P_{1} ⊕ P_{2} and let 1 = e_{1} + e_{2} be the corresponding orthogonal decomposition of the identity in End_{k}(P). Then End_{kJ}(P) is local, and the trivial group is a defect group of 1_{P} in J. Moreover, {e_{1}} and {e_{2}} are the only source points of the J-algebra End_{k}(P). These points are conjugate in J, and each has stabilizer I. Let ρ_{V} := ρ_{V,G}^{End_{k}(M),tr} and ρ_{1} := ρ_{1,J}^{End_{k}(P),tr} be Broué-Robinson bilinear forms. Applying (1) of Proposition (1.8) of [2] twice, first to ρ_{V} and then to ρ_{1}, we get

(7)  ρ_{V} (f_{1}, f_{2}) = ρ_{1}(θ(f_{1}), θ(f_{2})), for all f_{1}, f_{2} ∈ Tr_{V}^{G×Σ}(End_{kV} (M)).

Here θ is the composition End_{kV} (M) ↠ End_{k}(P_{1}) × End_{k}(P_{2}) ↪ End_{k}(P).

The group J × Σ acts on End_{k}(P_{1}) × End_{k}(P_{2}), with σ acting as an involutary anti-automorphism. In addition, e_{1}^{σ} = e_{2} and e_{2}^{σ} = e_{1}. We are in the situation of Theorem A.5 of our Appendix; there is a unique involutary k-algebra anti-automorphism σ̂ of End_{k}(P) whose restriction to End_{k}(P_{1}) × End_{k}(P_{2}) coincides with σ. Moreover, there exists a central extension H of J by a finite cyclic 2′-group Z and a commutative diagram of groups:

(8)  \begin{aligned}1 & ⟶ & Z & ⟶^{inc} & H & ⟶^{π} & J & ⟶ & 1 \\ & & ↓^{η} & & ↓^{τ} & & ↓^{ρ} \\ 1 & ⟶ & ∇(k) & ⟶^{inc} & C(σ̂) & ⟶^{π} & C(σ) & ⟶ & 1 \end{aligned}

In particular σ̂ is the adjoint of a nondegenerate H-invariant symmetric bilinear form b̂ on the kH-module P. For notational simplicity we will use σ for σ̂.

Theorem 7.2. Let M be an indecomposable kG-module that affords a nondegenerate G-invariant symmetric bilinear form. Let V be a vertex of M. Then there exists n ∈ N_{G}(V) with n^{2} ∈ V, such that End_{k}(M)↓_{\underline{G}×Σ} has a Scott component with vertex \underline{V} < \underline{n}σ > if and only if a source of M has odd dimension.

Proof. We keep the notation and assumptions of this section. In particular we assume that dim(S) is odd. We initially suppose that S ≅ S̸^{∗}. So [J : I] = 2. The restriction of P to the inverse image of I in H is a sum of P_{1} and its dual P_{2}. Thus P is not the projective cover of the trivial kH-module. However P is a self-dual principal indecomposable kH-module.

Set Bil_{k}(P)_{0} := P_{1}^{∗} ⊗ P_{2}^{∗} + P_{2}^{∗} ⊗ P_{1}^{∗} and Bil_{k}(P)_{1} := Bil_{k}(P_{1}) + Bil_{k}(P_{2}). Then Bil_{k}(P) = Bil_{k}(P)_{0} + Bil_{k}(P)_{1} is a direct sum decomposition as k\underline{H} × Σ-modules. As e_{1}^{σ} = e_{2} and e_{2}^{σ} = e_{1}, the form b̂ vanishes on P_{1} × P_{1} and also on P_{2} × P_{2}. Thus b̂ belongs to Bil_{k}(P)_{0}.

Identify P with ekH, where e is a primitive idempotent in kH. Lemma 6.2 implies that there exists a ∈ kH such that a is even and symmetric and b̂ agrees with the restriction of B_{a} to ekH. By Lemma 6.4, there exists t ∈ Ω(H) such that b̂(et ⊗ e) ≠ 0_{k}. But b̂ belongs to Bil_{k}(ekH)_{0}^{G×Σ}, while et ⊗ e belongs to (Bil_{k}(ekH)^{∗})^{<tσ>}. We conclude from remark 3.2 and Lemma 3.4 that Bil_{k}(P)_{0} has a Scott component with vertex < tσ >.

Recall that B : End_{k}(P) → Bil_{k}(P), such that B_{f}(u, v) := b̂(uf, v), for f ∈ End_{k}(P) and u, v ∈ P, is a H ≀Σ-module isomorphism. Under this isomorphism the \underline{H} × Σ-submodule End_{k}(P_{1}) + End_{k}(P_{2}) is mapped onto Bil_{k}(P)_{0}. So End_{k}(P_{1}) + End_{k}(P_{2}) has a Scott component with vertex < \underline{t}σ >, as \underline{H} × Σ-module. Let n be an element of N_{G}(V)/V whose image \overline{n} in N_{G}(V)/V coincides with the image of t in J = H/Z. In particular n^{2} ∈ V. Now Z is a normal 2′-subgroup of H that acts trivially on End_{k}(P_{1}) + End_{k}(P_{2}). It follows that End_{k}(P_{1}) + End_{k}(P_{2}) has a Scott component with vertex < \overline{n}σ >, as J × Σ-module.

The previous paragraph shows that there exist f_{1}, f_{2} ∈ End_{kG×Σ}(M) such that θ(f_{1}), θ(f_{2}) ∈ Tr_{<tσ>}^{J×Σ}(End_{k<tσ>}(P)) and ρ_{1}(θ(f_{1}), θ(f_{2})) ≠ 0_{k}. Since End_{k}(M) has vertex V ≀ Σ, as G ≀ Σ-module, we may write f_{1} = ∑_{u} f_{1u} and f_{2} = ∑_{u} f_{2u}, where u ranges over certain elements of N_{G}(V) with u^{2} ∈ V, and f_{1u}, f_{2u} ∈ Tr_{V <uσ>}^{G×Σ}(End_{kV <uσ>}(M)). Let \overline{u} denote the image of u in N_{G}(V)/V. Then

θ(Tr_{V <uσ>}^{G×Σ}(End_{kV <uσ>}(M))) ⊆ Tr_{<\overline{u}σ>}^{J×Σ}(End_{k<\overline{u}σ>}(P)).

Using Lemma 3.3 twice, we get

ρ_{1}(θ(f_{1n}), θ(f_{2n})) = ρ_{1}(θ(f_{1}), θ(f_{2n})) = ρ_{1}(θ(f_{1}), θ(f_{2})).

We deduce from this and Equation (7) that

ρ_{V} (f_{1n}, f_{2n}) ≠ 0_{k}.

But f_{1n}, f_{2n} ∈ Tr_{V <nσ>}^{G×Σ}(End_{kV <nσ>}(M)). We conclude from this and Proposition 3.5 that End_{k}(M)↓_{G×Σ} has a Scott component with vertex \underline{V} < \underline{n}σ >.

The arguments are simpler when S is self-dual and J = I. In particular we can reach the desired conclusion without appealing to Theorem A.5. We leave the details to the reader. □

8. Acknowledgement

The ideas in the first half of this paper originated from conversations with R. Gow and W. Willems, whilst the latter was visiting University College Dublin in January 2005.

Appendix A. Anti-automorphisms and G-algebras

The aim of this appendix is to prove Theorem A.5. This enables us to lift projective representations of a group in a way that is compatible with an involutary algebra anti-automorphism.

If A is a k-algebra, we let Aut(A) denote the group of all automorphisms of A and we let Aut^{∗}(A) denote the group of all automorphisms and anti-automorphisms of A. So each α ∈ Aut^{∗}(A) is a k-linear isomorphism of A such that either (ab)^{α} = a^{α}b^{α} for all a, b ∈ A, or (ab)^{α} = b^{α}a^{α} for all a, b ∈ A.

Fix an even dimensional k-vector space V and a decomposition V = V_{1} ⊕ V_{2}, where dim(V_{1}) = dim(V_{2}). Let 1_{E} = ϵ_{1} + ϵ_{2} be the corresponding orthogonal idempotent decomposition in E = End_{k}(V). Now ϵ_{i}Eϵ_{j} can be identified with

E_{ij} := Hom_{k}(V_{i}, V_{j}). In this way E has a matrix representation E = [\begin{matrix}E_{1} & E_{12} \\ E_{21} & E_{2} \end{matrix}],

where for notational simplicity E_{i} denotes E_{ii}.

The general linear group GL(V) of V is the group units in E. We identify GL(V_{1}) × GL(V_{2}) ≤ GL(V) with the set of elements g_{1} + g_{2} ∈ E such that g_{i} is a unit in E_{i}. The factor group PGL(V) = GL(V)/k^{×}1_{E} is naturally isomorphic to Aut(E). If θ(g) denotes the image of g ∈ GL(V) in Aut(E), then f^{θ(g)} = g^{−1}fg, for all f ∈ E.

Let N(ϵ_{1}, ϵ_{2}) denote the stabilizer subgroup of the set {ϵ_{1}, ϵ_{2}} in Aut(E), and let GL(V_{1}, V_{2}) be the inverse image of N(ϵ_{1}, ϵ_{2}) in GL(V). As V_{1} and V_{2} are isomorphic subspaces of V, there is a unit τ in E such that ϵ_{1}τ = τϵ_{2}. Replacing τ by ϵ_{1}τ + τ^{−1}ϵ_{1}, we can and do assume that τ is an involution. It is clear that GL(V_{1}, V_{2}) = GL(V_{1}) × GL(V_{2}) :< τ >, a group that is isomorphic to GL_{d}(k) ≀ Σ_{2}.

Restriction to E_{1} ×E_{2} induces a group homomorphism ϕ : N(ϵ_{1}, ϵ_{2}) → Aut(E_{1} × E_{2}). Each α ∈ Aut(E_{1} × E_{2}) satisfies ϵ_{i}^{α} ∈ {ϵ_{1}, ϵ_{2}}, for i = 1, 2. If ϵ_{i}^{α} = ϵ_{3−i} then ϵ_{i}^{αt} = ϵ_{i}, while if ϵ_{i}^{α} = ϵ_{i} then we can identify α, via its restrictions to E_{1} and to E_{2}, with an element of Aut(E_{1}) × Aut(E_{2}). It follows that Aut(E_{1} × E_{2}) = Aut(E_{1}) × Aut(E_{2}) :< ϕ(τ) >, a group that is isomorphic to PGL_{d}(k) ≀ Σ_{2}.

Our lemma is a consequence of this discussion:

Lemma A.1. Every k-automorphism of E_{1}×E_{2} extends to an inner automorphism of E. The kernel of the surjective map ϕθ : GL(V_{1}, V_{2}) → Aut(E_{1} × E_{2}) is k^{×}ϵ_{1} + k^{×}ϵ_{2}.

We now discuss k-algebra anti-automorphisms. Fix a non-degenerate symmetric bilinear k-form b_{1} on V_{1}. Then Q(v) := b_{1}(vϵ_{1}, vϵ_{2}τ), for v ∈ V, defines a quadratic form on V. Let b be the polarization of Q. So b(u, v) = b_{1}(uϵ_{1}, vϵ_{2}τ)+b_{1}(vϵ_{1}, uϵ_{2}τ), for all u, v ∈ V. The adjoint of b is an involution β ∈ Aut^{∗}(E)\Aut(E) such that τ^{β} = τ and ϵ_{1}^{β} = ϵ_{2} and ϵ_{2}^{β} = ϵ_{1}. Also Aut^{∗}(E) = Aut(E) :< β >, as the product of two anti-automorphisms is an automorphism.

Let g ∈ GL(V) and f ∈ E. Then f^{βθ(g)β} = (g^{−1}f^{β}g)^{β} = g^{β}fg^{−β}. So

(9)  θ(g)^{β} = θ(g^{−β}) in Aut(E).

For instance, θ(τ)^{β} = θ(τ), as τ is an involution.

Let N^{∗}(ϵ_{1}, ϵ_{2}) be the stabilizer subgroup of the set {ϵ_{1}, ϵ_{2}} in Aut^{∗}(E). Then β belongs to N^{∗}(ϵ_{1}, ϵ_{2})\N(ϵ_{1}, ϵ_{2}). Restriction gives a group homomorphism, also denoted by ϕ, from N^{∗}(ϵ_{1}, ϵ_{2}) into Aut^{∗}(E_{1}×E_{2}). Clearly N^{∗}(ϵ_{1}, ϵ_{2}) = N(ϵ_{1}, ϵ_{2}) :< ϕ(β) > and

Aut^{∗}(E_{1} ×E_{2}) = Aut(E_{1} ×E_{2}) :< ϕ(β) >= Aut(E_{1})×Aut(E_{2}) :< ϕ(β), ϕ(τ) > .

The latter group is isomorphic to Aut^{∗}(E_{1}) ≀ Σ_{2} and also to a group PGL_{d}(k)^{2} : ℤ_{2}^{2}. We summarise this discussion with:

Lemma A.2. Every k-algebra anti-automorphism of E_{1} × E_{2} can be extended to a k-algebra anti-automorphism of E. The extensions of a single anti-automorphism form a coset of θ(k^{×}ϵ_{1} + k^{×}ϵ_{2}) in N^{∗}(ϵ_{1}, ϵ_{2}).

For involutions in Aut^{∗}(E_{1} × E_{2}), we even have:

Lemma A.3. Let σ be an involutary k-algebra anti-automorphism of E_{1} × E_{2} such that ϵ_{1}^{σ} = ϵ_{2}. Then there is a unique extension of σ to an involutary antiautomorphism σ̂ of E.

Proof. Let α be any element of N^{∗}(ϵ_{1}, ϵ_{2}) satisfying ϕ(α) = σ. Then αβ is a k-algebra automorphism of E_{1} × E_{2} and moreover ϵ_{i}^{αβ} = ϵ_{i}, for i = 1, 2. So α = θ(g_{1} + g_{2})β, for some units g_{i} ∈ E_{i}. Also {α_{μ} := θ(μg_{1} + g_{2})β | μ ∈ k^{×}} is the set of extensions of σ to E.

Let us denote the inverse of g_{i} in E_{i} by g_{i}^{−1}. As ϵ_{i}^{β} = ϵ_{3−1} and β is an algebra anti-automorphism, we have (g_{i}^{−1})^{β} = (g_{i}^{β})^{−1} in E_{3−i}. We write g_{i}^{−β} for this common element. For μ ∈ k^{×}, we see from (9) that

α_{μ}^{2} = θ(μg_{1} + g_{2})θ(μg_{1} + g_{2})^{β} = θ(μg_{1}g_{2}^{−β} + μ^{−1}g_{2}g_{1}^{−β}).

As σ is an involution, α^{2} acts as the identity on both E_{1} and E_{2}. In particular g_{1}g_{2}^{−β} = λϵ_{1}, for some λ ∈ k^{×}. It follows that g_{2}^{−β} is a scalar multiple of g_{1}^{−1}, whence g_{2}^{−β} commutes with g_{1}. Thus g_{2}^{−β}g_{1} = λϵ_{1}. Applying ^{−β} to this, we deduce that g_{2}g_{1}^{−β} = λ^{−1}ϵ_{2}. Thus α_{μ}^{2} = θ(μλϵ_{1} + μ^{−1}λ^{−1}ϵ_{2}).

The last paragraph implies that the extension α_{μ} is an involution in Aut^{∗}(E) if and only if μλ = μ^{−1}λ^{−1}, which holds if and only if μ = λ^{−1}. We conclude that σ̂ := α_{λ^{−1}} is the unique extension of σ to E that is an involution. □

Fix an involutary k-algebra anti-automorphism σ of E_{1} × E_{2} such that ϵ_{1}^{σ} = ϵ_{2}. Denote by σ̂ the unique involution in N^{∗}(ϵ_{1}, ϵ_{2}) such that ϕ(σ̂) = σ. Let C(σ) denote the centralizer of σ in Aut(E_{1} × E_{2}) and define

C(σ̂) := {g ∈ GL(V_{1}, V_{2}) | gσ̂ = g^{−1}}.

As σ̂ is an anti-automorphism, C(σ̂) is a subgroup of GL(V_{1}, V_{2}). Note that if g ∈ C(σ̂), then θ(g) commutes with σ̂, and hence ϕθ(g) belongs to C(σ).

Lemma A.4. The map ϕθ induces a group epimorphism C(σ̂) ↠ C(σ).

Proof. Let x ∈ C(σ). Choose g ∈ GL(V_{1}, V_{2}) such that ϕθ(g) = x. Then ϕθ(ggσ̂) = ϕθ(g)ϕθ(g^{−1})^{σ} = 1. It follows that ggσ̂ = λ_{1}ϵ_{1} + λ_{2}ϵ_{2}, for some λ_{1}, λ_{2} ∈ k^{×}. But σ̂ is an involutary k-algebra anti-automorphism. So ggσ̂ is fixed by σ̂. Applying σ̂ to λ_{1}ϵ_{1} + λ_{2}ϵ_{2} we see that λ_{1} = λ_{2}. We deduce from this that gσ̂ = λ_{1}g^{−1}. As k is perfect and has characteristic 2, there exists μ ∈ k^{×} such that μλ_{1} = μ^{−1}. Then

(μg)σ̂ = μg^{s} = μλ_{1}g^{−1} = (μg)^{−1}.

So μg ∈ C(σ̂), which completes the proof. □

Set ∇(k) := {(λ, λ^{−1}) ∈ GL(V_{1}) × GL(V_{2})}, a subgroup of GL(V). So ∇(k) is the kernel of the restriction of ϕθ to C(σ̂). We now give the main result of this section.

Theorem A.5. Let V, E, E_{i}, ϵ_{i} be as above and let σ be an involutary antiautomorphism of E_{1} × E_{2} such that ϵ_{1}^{σ} = ϵ_{2}, and let σ̂ be the unique involutary anti-automorphism of E whose restriction to E_{1} × E_{2} coincides with σ. Suppose that ρ : G → C(σ) is a group homomorphism. Then there is a commutative diagram of groups

(10)  \begin{aligned}1 & ⟶ & Z & ⟶^{inc} & Ĝ & ⟶^{π} & G & ⟶ & 1 \\ & & ↓^{η} & & ↓^{τ} & & ↓^{ρ} \\ 1 & ⟶ & ∇(k) & ⟶^{inc} & C(σ̂) & ⟶^{θ} & C(σ) & ⟶ & 1 \end{aligned}

Here Ĝ is a finite central extension of G by a cyclic group Z of odd order. In particular, σ̂ is the adjoint of a nondegenerate Ĝ-invariant symmetric bilinear form on V.

Proof. This is a consequence of Lemma A.4 and standard arguments involving pull-back diagrams and cohomology. One could combine Proposition (10.5) and the methods of Example (10.8) in [11], for instance. □

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Mathematics Department, National University of Ireland, Maynooth, Co. Kildare, Ireland.

  E-mail address: John.Murray@maths.nuim.ie