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
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