Finitely asymptotic properties of powers of characters of compact semisimple Lie groups.

Abstract

This thesis has to do with decomposing tensor powers of irreducible representations of compact semisimple Lie groups or their Lie algebras. We will be concerned only with the set of irreducible representations appearing in the decomposition. In particular, we determine whether, for a sufficiently high tensor power, this set is "maximal", in a sense to be made more precise below. We rely on the theory of semisimple Lie algebras, and describe the results in terms of characters. A (reducible) character of a finite dimensional complex semisimple Lie algebra is saturated if for each of its dominant weights, the corresponding irreducible character is a summand in its orthogonal decomposition. A character is eventually saturated if some power is saturated. In the first part of the thesis, we describe all eventually saturated irreducible characters. In particular, it is shown that any irreducible character whose highest weight is in the interior of the Weyl chamber is eventually saturated; if the Lie algebra is simply laced, then all irreducible characters are eventually saturated; if not, then there are irreducible characters (with highest weights on the boundary of the Weyl chamber) that are not eventually saturated. We use the PRV conjecture to derive necessary and sufficient conditions for eventual saturation. These conditions are expressed in terms of cones generated by the weights of the character. The convex hull of the weight diagram, and the weights adjacent to the highest weight along the edges of the convex hull, are described in detail. We show in the second part of the thesis that if the Lie algebra is A$\sb{d}$, and $d \ \le$ 5, then for any integer $n \ge \ d$ + 1 and any irreducible character $\chi\lambda$, the product $\chi\sbsp{\lambda}{n}$ is saturated. We also establish this result for certain irreducible characters of $A\sb{d}, d \ge$ 5. These results are proved by induction on the rank of the Lie algebra and on the highest weight of the character. The geometry of the convex hull of the set of weights comes in to play here as well, and the dominant faces of this set are described. A reduction result, relating the "restriction to a dominant face" of the decomposition of a product of characters to that of a corresponding product in an algebra of lower rank, is established. The Littlewood-Richardson rule is used to compare products in which the component irreducible characters have highest weights which differ by a small amount. Similar induction arguments are used to describe all the irreducible characters appearing in the decomposition of a product of irreducible characters of A$\sb2$. Some of the irreducible characters appearing in a product of characters of higher rank algebras can also be determined using this type of induction. The questions considered here arise in the study of product type actions of compact groups. The results on eventual saturation of irreducible characters are useful in computing the equivariant ordered K-theory of certain fixed point C$\sp*$-algebras under the corresponding group actions.