4 1. INTRODUCTION
p|H0 extends to a representation of H such that V |H is irreducible, but V |H0 is re-
ducible.
Given a triple (G, H, V ) satisfying Hypothesis 1, let λ and δ denote the highest
weights of the KG-module V and the
KH0-module
W , respectively (see Section
2.1 for further details). Note that W is self-dual as a
KH0-module
(the condition
H GL(W ) implies that the relevant graph automorphism of
H0
acts on W ),
so
H0
fixes a non-degenerate form on W and thus G is either a symplectic or
orthogonal group (by condition S3 above). We write λ|H0 to denote the restriction
of the highest weight λ to a suitable maximal torus of
H0.
Remark 2. If (G, H, V ) is a triple satisfying the conditions in Hypothesis 1
then either H G, or G = Dn and H Dn.2 = GO(W ). In Theorem 3 below we
describe all the irreducible triples (G, H, V ) satisfying Hypothesis 1. By inspecting
the list of examples with G = Dn we can determine the cases with H G, and this
provides a complete classification of the relevant triples with H in the collection S
of subgroups of G.
Remark 3. If (G, p) = (Bn, 2) then G is reducible on the natural KG-module
W (the corresponding symmetric form on W has a 1-dimensional radical). In
particular, no positive-dimensional subgroup H of G satisfies condition S2 above, so
in this paper we will always assume p = 2 when G = Bn. (The only exception to this
rule arises in the statement of Theorems 4 and 5, where we allow (G, p) = (Bn, 2).)
Theorem 1. A triple (G, H, V ) with H G satisfies Hypothesis 1 if and only
if (G, H, λ, δ) = (C10,A5.2,λ3,δ3), p = 2, 3 and λ|H0 = δ1 + 2δ4 or 2δ2 + δ5.
Remark 4. Note that in the one example that arises here, V |H0 has p-restricted
composition factors. However, this is a new example, which is missing from Ford’s
tables in [8]. See Remark 3.6.18 for further details. It is also important to note
that the proof of Theorem 1 is independent of Ford’s work [8, 9]; our analysis
provides an alternative proof (and correction), without imposing any conditions on
the composition factors of V |H0 .
More generally, using [23, Theorem 2], we can determine the triples (G, H, V )
satisfying the following weaker hypothesis:
Hypothesis 2. G and H are given as in Hypothesis 1, V is a rational tensor
indecomposable p-restricted irreducible KG-module such that V |H
is irreducible,
and V is not the natural KG-module, nor its dual.
Theorem 2. The triples (G, H, V ) with H G satisfying Hypothesis 2 are
listed in Table 1.
Remark 5. Let us make some remarks on the statement of Theorem 2:
(a) In Table 1, κ denotes the number of
KH0-composition
factors of V |H0 .
(b) Note that V |H0 is irreducible in each of the cases listed in the final four
rows of Table 1; these are the cases labelled II1, S1, S7 and S8, respectively,
in [23, Table 1].
(c) Note that A3.2 D7 and D4.S3 D13 (see Theorem 2.5.1), so the cases
listed in the final two rows of Table 1 give rise to genuine examples with
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