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