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 , Guralnick and Tiep consider irreducible triples (G, H, V ) in the special case
G = SL(W ) with V =
), 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
) is in progress. These results have found interesting
applications in the study of holonomy groups of stable vector bundles on smooth
projective varieties (see ). We also refer the reader to  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 .
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
) 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  on irreducibly acting connected subgroups, we use a result of Smith ,
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