Classical Compact Groups Research Articles

Spectrum of Casimir Invariants for the Simple Classical Lie Groups

The spectra of the Casimir invariants for the classical compact simple Lie groups are presented. It is proved that these invariants are irreducible and functionally independent. The highest weight of a representation is determined in terms of its invariant spectrum.

Inner and Restriction Multiplicity for Classical Groups

For the classical compact Lie groups G, a formula for the multiplicity of weights (called inner multiplicity) is given. This formula relates the inner multiplicity of a group G to the inner multiplicity of a naturally embedded subgroup G′. For the SU(n) groups the formula can be brought into a particularly simple form—namely, a sum over Kronecker symbols—by choosing the group SU(2) for G′. The multiplicity of irreducible representations of a subgroup G′ into which an irreducible representation of a group G decomposes if G is restricted to G′—called restriction multiplicity of G with respect to G′—is related to the inner multiplicity of the group G.

The cohomology of quotients of classical groups

THE HOMOLOGY and cohomology rings of the classical compact Lie groups so(n), SU(n), Sp(n) are well known (for example see Bore1 [3]). Most of these groups have non-trivial centers, and in [4], Bore1 investigated the quotients of these groups by central subgroups, in particular, calculating cohomology rings with Zp coefficients, p prime. In this paper we pursue the investigation of these quotients further, extending Borel’s results. We obtain extra information on the integral cohomology and we completely determine the diagonal maps in cohomology with 2, coefficients p prime, and the action of the Steenrod algebra in any quotient of one of these groups by a central subgroup. This information leads to some applications such as the fact that homotopy equivalent compact connected simple groups are isomorphic, and a technique to prove facts about vector fields on real projective spaces, (see $9). These results on vector fields may be applied to prove non-immersion theorems for real projective spaces. In particular, it is shown that if n = 2’ + 3, r 2 3, then P*-l and P” do not immerse in R2”-7. Mahowald [9] and Sanderson [12] have shown that P” does immerse in RZn-‘, so that this is the best possible immersion for P” and P”-‘. The diagonal maps in the cohomology of these quotient groups with Z, coefficients are usually non-commutative, where p divides the order of the central subgroup, the only exceptions being low dimensional and PSp(n) for odd n (mod 2), (where PG denotes the quotient of G by its center). In particular ifp is an odd prime and p divides the order of the central subgroup, then the diagonal map in cohomology modp is non-commutative. Araki [2] has shown that the exceptional groups which have 3-torsion in homology have non-commutative diagonal maps in cohomology mod 3. One may conjecture: If a Lie group G hasp-torsion in its homology (p an odd prime) then the diagonal map in H*(G; Z,) is not commutative.

An application of the Morse theory to the topology of Lie-groups

2. The Freudenthal theorem for symmetric spaces Our primary interest is in the classical compact groups; nevertheless, it is essential for our method to consider the larger family of compact symmetric spaces. A compact homogeneous Riemannian manifold M is called symmetric if it admits an 'inverse operation', i.e. if M admits an isometry, keeping a point P e M fixed, and whose differential at P is — 1. These geometric generalizations of the compact groups seem to be the class of spaces to which the Morse theory is most applicable. The reason is that, on such a space, conjugate points have global implications: