Remark In (ii) of the theorem, the possibilities for
are given by Theorem
5, and those for Z by [Se2, Theorem 1], They are:
Y0 possibilities for Z CG(Y0) (G = Es, E7, E6j F4)
F4 Au A1G29 G2 G2, Au 1, l(resp.)
E& A2, G2l A2G2, C4, F4 A2, Ti, 1, -
E7 Ai, A2j AXAU A1G2, A1F4, G2C3 Au 1, - , -
£^8 A\, B2, AiA2, G2F4 1? - , ? -
In Section 7 we also define the labelled diagram of an arbitrary closed connected
simple subgroup X of G, and prove the following result, which comes close to showing
that the labelled diagram determines X up to (Aut G)-conjugacy.
Theorem 8 Let X\,X2 he closed connected simple subgroups of G of the same type,
and of rank at least 2. Assume that X\ and X2 have the same labelled diagram, and
that p = 0 or p 7V(Xi, G). Then with two exceptions, there is a subsystem subgroup
Y of G such that
(i) Y contains X\ and a conjugate of X2,
(ii) Y is minimal subject to containing X\, and
(Hi) X\ and X% are conjugate in Aut Y.
In the exceptions, G = E7 (resp. Es), Xi = X2 = = A2 and X\^X2 lie in subsystem
subgroups A5 and
(resp. A2A5 and D4D4).
In the last sentence of Theorem 8, A$ and A$ are representatives of the two
conjugacy classes of subsystem subgroups of type A5 in E7.
We use the following notation throughout the paper. If X is a connected reductive
group over the algebraically closed field K, and A is a dominant weight, then Vx(A)
denotes the rational irreducible iTX-module with high weight A, and Wx(^) denotes
the corresponding Weyl module. If Vi,..., Vk are modules, then V\j ... /14 denotes
a module having the same composition factors as Vi © .. Vfc. We often abbreviate
this notation slightly as follows: if/xi,... ,/ifc are dominant weights for the group X,
and ci,.. . ,cjt are positive integers, then
always denotes a A'X-module having the same composition factors as
When X is a subgroup of G and V is a A'G-module, we use V j X for the
restriction of V to X. If the characteristic p 0, and q is a power of p, then aq
denotes a standard Frobenius morphism of X - that is, a morphism inducing the
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