INVARIANT RINGS AND DUAL REPRESENTATIONS OF DIHEDRAL GROUPS

Abstract. The Weyl group of a compact connected Lie group is a reflec-tion group. If such Lie groups are locally isomorphic, the representations of the Weyl groups are rationally equivalent. They need not however be equivalent as integral representations. Turning to the invariant the-ory, the rational cohomology of a classifying space is a ring of invariants, which is a polynomial ring. In the modular case, we will ask if rings of invariants are polynomial algebras, and if each of them can be realized as the mod p cohomology of a space, particularly for dihedral groups. Suppose G is a compact connected Lie group. The Weyl group W (G) acts on a maximal torus Tn, and the integral representation W (G)−→GL(n,Z) obtained makes W (G) a reflection group. Recall that if such Lie groups are locally isomorphic, the representations of the Weyl groups are equivalent over Q. They need not however be equivalent as Z-representations. For instance, the integral representation of W (PU(n)) is not equivalent to that of W (SU(n)). LetW (G) ∗ denote the dual representation ofW (G). Then we seeW (PU(n)) = W (SU(n)) ∗ [11]. Turning to the rings of invariants, for the cohomology of classifying spaces BG, it is well-known that H∗(BG;Q) = H∗(BTn;Q)W (G), which is a poly-nomial ring. We recall that Q can be replaced by a finite field Fp when the prime p is large. Here W is a pseudoreflection subgroup of GL(n,Fp), [21, Ch 7] and [13, Part VI]. If the order of W is prime to p, according to [7, The-orem 1.5 and Lemma 5.2], we see H∗(BTn;Fp)W ∼ = H∗(BTn;Fp)W ∗. So we will consider mainly the case that |W | ≡ 0 mod p. For dihedral groups and symmetric groups, we will consider dual representations, invariant rings and the realizability in the modular case. We start with an example H∗(BTn;Fp)W (G) that is not isomorphic to the ring of invariants by the dual representation H∗(BTn;Fp)W (G). Recall that