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

On Fusion Algebras and Modular Matrices

We consider the fusion algebras arising in e.g. Wess-Zumino-Witten conformal field theories, affine Kac-Moody algebras at positive integer level, and quantum groups at roots of unity. Using properties of the modular matrix $S$, we find small sets of primary fields (equivalently, sets of highest weights) which can be identified with the variables of a polynomial realization of the $A_r$ fusion algebra at level $k$. We prove that for many choices of rank $r$ and level $k$, the number of these variables is the minimum possible, and we conjecture that it is in fact minimal for most $r$ and $k$. We also find new, systematic sources of zeros in the modular matrix $S$. In addition, we obtain a formula relating the entries of $S$ at fixed points, to entries of $S$ at smaller ranks and levels. Finally, we identify the number fields generated over the rationals by the entries of $S$, and by the fusion (Verlinde) eigenvalues.Comment: 28 pages, plain Te

On Robin boundary conditions and the Morse potential in quantum mechanics

The physical origin is investigated of Robin boundary conditions for wave functions at an infinite reflecting wall. We consider both Schr\"odinger and phase-space quantum mechanics (a.k.a. deformation quantization), for this simple example of a contact interaction. A non-relativistic particle moving freely on the half-line is treated as moving on the full line in the presence of an infinite potential wall, realized as a limit of a Morse potential. We show that the wave functions for the Morse states can become those for a free particle on the half-line with Robin boundary conditions. However, Dirichlet boundary conditions (standard walls) are obtained unless a mass-dependent fine tuning (to a reflection resonance) is imposed. This phenomenon was already observed for piece-wise flat potentials, so it is not removed by smoothing. We argue that it explains why standard quantum walls are standard. Next we consider the Wigner functions (the symbols of both diagonal and off-diagonal density operator elements) of phase-space quantum mechanics. Taking the (fine-tuned) limit, we show that our Wigner functions do reduce to the expected ones on the half-line. This confirms that the Wigner transform should indeed be unmodified for this contact interaction.Comment: 19 page

Lithic economies and community organization at La Laguna, Tlaxcala

Site-wide, assemblage-based lithic analyses help to elucidate community dynamics including variability in domestic economies, technological skill and decision making, exchange networks, and ritual practices. In this study we present the results of an analysis of over 36,000 lithic artifacts from the site of La Laguna, Tlaxcala. We compare Middle to Late Formative period (ca. 600–400 b.c.) and Terminal Formative period (ca. 100 b.c.–a.d. 150) deposits to examine transformations associated with urbanization and state formation during this interval. The residents of La Laguna had relatively equal and ample access to obsidian, and most production was organized independently by households. We identify blade production zones and variability in consumption patterns suggestive of different domestic, communal, and ceremonial activities. The introduction of bloodletters, elaborate large bifacial knives, and zoomorphic eccentrics to the Terminal Formative assemblage may indicate the emergence of higher statuses, new social roles, and militaristic symbolism during this period