2. Labelled Diagrams In this section we shall develop machinery which will be needed throughout the remainder of this work. This enables us to associate a labelled diagram to an A\ subgroup of an algebraic group G according to [17, Theorem 6] this labelled diagram determines the (Aut G)-conjugacy class of X, if G is of exceptional type and is defined over an algebraically closed field of characteristic greater than N(Ai,G). We shall obtain an algorithm for transforming a labelled diagram into a canonical form in addition we shall see how to pass between the torus of AT, its labelled diagram and the set of weights in the action of X on C(G). We begin by considering labelled diagrams in the abstract, removed from the context of A\ subgroups. Let $ be a root system of rank L DEFINITION. A labelling is a map A : £ Z which is the restriction to $ of a linear map Z$ - Z thus if a,/?,* + /? e $ then A(a + /?) = A(a) + A(/?). A labelled diagram is a Dynkin diagram with an integer attached to each node the integers are called labels. We write V = £(!) for the set of labelled diagrams, which may be given the structure of an abelian group by pointwise addition thus V = J}. Given a simple system II = {ai,... , a^} (which we regard as an ordered set), there is a bijective correspondence between the set of labellings and £, under which the labelling A corresponds to the labelled diagram whose zth node has label A(c^). We call this labelled diagram A(A II). We have an action of Aut3 on labellings, given by (OX)(a) = A(#_1a) for 0 G Aut £ . This carries over to an action on labelled diagrams, by setting #A(A II) = A(0A II) = A(A 0_1II). Hence V is partitioned into (Aut $)-orbits since Aut I = W.T by [11, 12.2], where V is the group of graph automorphisms of £, each (Aut 3)- orbit is a union of V^-orbits. DEFINITION. A non-negative labelled diagram is a labelled diagram all of whose labels are non-negative. We write V+ = V+($) for the set of non-negative labelled diagrams thus £+ ^ N0e. It is shown in [17, Proposition 6.2] that each VF-orbit in V meets V+ exactly once (the result is stated only for labelled diagrams arising from A\ subgroups in the manner explained below, but the proof works for arbitrary labelled diagrams). Thus by considering the (Aut I)-orbit of the corresponding labelled diagram, we see that each labelling A has associated to it a non-negative labelled diagram, which is unique up to graph automorphisms. It is worth recording the following result. LEMMA 2.1. Let A e V and {A+} = W.A n £+ then if 7 e T such that 7A = A, we have 7A + = A + . PROOF. The 7-orbits in $ form a root system $1, and there are natural iso- morphisms £ : V^ - V($i) and $ : W1 - W($i) satisfying £(wA ) = ^(w)^(A') 5
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