6 1. INTRODUCTION

Moreover, for (G, H, V ) in [23, Theorem 2], [6, Table 1] or Table 1, H acts irre-

ducibly on V .

Similar problems have been studied recently by various authors. For instance,

in [13], Guralnick and Tiep consider irreducible triples (G, H, V ) in the special case

G = SL(W ) with V =

Sk(W

), the k-th symmetric power of the natural module

W for G, and H is any (possibly finite) closed subgroup of G. A similar analysis

of the exterior powers

Λk(W

) is in progress. These results have found interesting

applications in the study of holonomy groups of stable vector bundles on smooth

projective varieties (see [2]). We also refer the reader to [12] for related results on

the irreducibility of subgroups acting on small tensor powers of the natural module.

At the level of finite groups, a similar problem for subgroups of GLn(q) is studied

by Kleshchev and Tiep in [16].

As a special case of Theorem 4, we determine the maximal positive-dimensional

closed subgroups of a simple classical algebraic group G acting irreducibly on all

KG-composition factors of a symmetric or exterior power of the natural KG-

module. (The proof of Theorem 5 is given in Chapter 7.)

Theorem 5. Let G be a simple classical algebraic group over an algebraically

closed field K of characteristic p ≥ 0 and let H be a closed positive-dimensional

maximal subgroup of G. Let W be the natural KG-module and let n denote the

rank of G.

(i) Suppose 1 k n. Then H acts irreducibly on all KG-composition

factors of Λk(W ) if and only if (G, H, k) is one of the cases in Table 3.

(ii) Suppose 1 k, and k p if p = 0. Then H acts irreducibly on all KG-

composition factors of

Sk(W

) if and only if (G, H, k) is one of the cases

in Table 4.

Remark 7. In the third column of Tables 3 and 4 we describe the embedding

of H in G in terms of a suitable set of fundamental dominant weights for H0. In

addition, Tn denotes a maximal torus of dimension n in the third line of Table 3.

To close this introductory chapter, we would like to comment briefly on our

methods, and how they relate to those used by Ford in [8, 9]. As in the work of

Seitz [23] on irreducibly acting connected subgroups, we use a result of Smith [24],

which states that if P = QL is a parabolic subgroup of a semisimple algebraic group

G, and V is an irreducible KG-module, then L acts irreducibly on the commutator

quotient V/[V, Q]. In our set-up, this can be applied to the irreducible KG-module

V = VG(λ), as well as to the irreducible summands of V |X , where X = H0. In

order to exploit this property, for any parabolic subgroup PX = QX LX of X we

will construct a canonical parabolic subgroup P = QL of G as the stabilizer of the

sequence of subspaces

W [W, QX ] [[W, QX ],QX ] · · · 0 (1.1)

It turns out that the embedding PX P has several important properties (see

Lemma 2.7.1). For instance, LX is contained in a Levi factor L of P .

We can study the weights occurring in each of the subspaces in the above flag

of W to obtain a lower bound on the dimensions of the quotients, which then leads

to structural information on L . Moreover, we can use this to impose conditions on