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

Purely affine elementary su(N) fusions

We consider three-point couplings in simple Lie algebras -- singlets in triple tensor products of their integrable highest weight representations. A coupling can be expressed as a linear combination of products of finitely many elementary couplings. This carries over to affine fusion, the fusion of Wess-Zumino-Witten conformal field theories, where the expressions are in terms of elementary fusions. In the case of su(4) it has been observed that there is a purely affine elementary fusion, i.e., an elementary fusion that is not an elementary coupling. In this note we show by construction that there is at least one purely affine elementary fusion associated to every su(N>3).Comment: 9 pages, LaTe

Generalized Uncertainty Principle Corrections to the Simple Harmonic Oscillator in Phase Space

We compute Wigner functions for the harmonic oscillator including corrections from generalized uncertainty principles (GUPs), and study the corresponding marginal probability densities and other properties. We show that the GUP corrections to the Wigner functions can be significant, and comment on their potential measurability in the laboratory.Comment: minor revisions; included journal referenc

Zinc calixarene complexes for the ring opening polymerization of cyclic esters

Reaction of Zn(C₆F₅)₂·toluene (two equivalents) with 1,3-dipropoxy-p-tert-butyl-calix[4]arene (L¹H₂) led to the isolation of the complex [{Zn(C₆F₅)}₂L¹] (1), whilst similar use of Zn(Me)₂ resulted in the known complex [{Zn(Me)}₂L¹] (2). Treatment of L¹H₂ with in situ prepared Zn{N(SiMe₃)₂}₂ in refluxing toluene led to the isolation of the compound [(Na)ZnN(SiMe₃)₂L¹] (3). The stepwise reaction of L¹H₂ and sodium hydride, followed by ZnCl₂ and finally NaN(SiMe₃)₂ yielded the compound [Zn{N(SiMe₃)₂}₂L¹] (4). The reaction between three equivalents of Zn(C₆F₅)₂·toluene and oxacalix[3]arene (L²H₃) at room temperature formed the compound {[Zn(C₆F₅)]₃L²} (5); heating of 5 in acetonitrile caused the ring opening of the parent oxacalix[3]arene and rearrangement to afford the complex [(L²)Zn₆(C₆F₅)(R)(RH)OH]·5MeCN R = C₆F₅CH₂-(p-ᵗBuPhenolate-CH₂OCH₂–)₂–p-ᵗBuPhenolate-CH₂O⁻)³⁻ (6). The molecular structures of the new complexes 1, 3 and 6, together with that of the known complex 2, whose solid state structure has not previously been reported, have been determined. Compounds 1, 3–5 have been screened for the ring opening polymerization (ROP) of ε-caprolactone (ε-CL) and rac-lactide. Compounds featuring a Zn–C₆F₅ fragment were found to be poor ROP pre-catalysts as they did not react with benzyl alcohol to form an alkoxide. By contrast, compound 4, which contains a zinc silylamide linkage, was the most active of the zinc-based calix[4]arene compounds screened and was capable of ROP at ambient temperature with 65% conversion over 4 h