Polytope Expansion of Lie Characters and Applications

The weight systems of finite-dimensional representations of complex, simple Lie algebras exhibit patterns beyond Weyl-group symmetry. These patterns occur because weight systems can be decomposed into lattice polytopes in a natural way. Since lattice polytopes are relatively simple, this decomposition is useful, in addition to being more economical than the decomposition into single weights. An expansion of characters into polytope sums follows from the polytope decomposition of weight systems. We study this polytope expansion here. A new, general formula is given for the polytope sums involved. The combinatorics of the polytope expansion are analyzed; we point out that they are reduced from those of the Weyl character formula (described by the Kostant partition function) in an optimal way. We also show that the weight multiplicities can be found easily from the polytope multiplicities, indicating explicitly the equivalence of the two descriptions. Finally, we demonstrate the utility of the polytope expansion by showing how polytope multiplicities can be used in the calculation of tensor product decompositions, and subalgebra branching rules.Comment: 16 pages incl. 3 figures, to appear in J. Math. Phy

Demazure Characters and Affine Fusion Rules

The Demazure character formula is applied to the Verlinde formula for affine fusion rules. We follow Littelmann's derivation of a generalized Littlewood-Richardson rule from Demazure characters. A combinatorial rule for affine fusions does not result, however. Only a modified version of the Littlewood-Richardson rule is obtained that computes an (old) upper bound on the fusion coefficients of affine $A_r$ algebras. We argue that this is because the characters of simple Lie algebras appear in this treatment, instead of the corresponding affine characters. The Bruhat order on the affine Weyl group must be implicated in any combinatorial rule for affine fusions; the Bruhat order on subgroups of this group (such as the finite Weyl group) does not suffice.Comment: 23 pages, TeX, uses harvma

Bigger, faster, stronger! An overview of anabolic androgenic steroids and their use and impact on the sport industry

The use of anabolic androgenic steroids (AAS) in sport is no longer confined to the power disciplines and has become a wide-spread issue throughout the general population. AAS are synthetic versions of the male hormone testosterone and display both anabolic and androgenic properties. It is the anabolic properties that are responsible for the muscle binding characteristics and are the main attraction for users. The primary purpose of this review was to provide an overview of the use of AAS in the sports industry by outlining the history of AAS use, the role of AAS in the Olympic success of Soviet athletes and the German Democratic Republic. Furthermore, case studies of the high profile cases of Ben Johnson, Marion Jones and Lance Armstrong were also examined along with the consequences of their drug use. Also outlined are the reasons for AAS use, the variety of ways in which they are used and short and long-term adverse side effects associated with their use. This research has highlighted problems with previous AAS literature as there is a lack of research into the long-term side effects of AAS use