Crystal duality and Littlewood-Richardson rule of extremal weight crystals

We consider a category of \gl_\infty-crystals, whose objects are disjoint unions of extremal weight crystals of non-negative level with certain finite conditions on the multiplicity of connected components. We show that it is a monoidal category under tensor product of crystals and the associated Grothendieck ring is anti-isomorphic to an Ore extension of the character ring of integrable lowest weight \gl_\infty-modules with respect to derivations shifting the characters of fundamental modules. A Littlewood-Richardson rule of extremal weight crystals with non-negative level is described explicitly in terms of classical Littlewood-Richardson coefficients

Heat capacity of α\alpha-GaN: Isotope Effects

Until recently, the heat capacity of GaN had only been measured for polycrystalline powder samples. Semiempirical as well as \textit{first-principles} calculations have appeared within the past few years. We present in this article measurements of the heat capacity of hexagonal single crystals of GaN in the 20-1400K temperature range. We find that our data deviate significantly from the literature values for polycrystalline materials. The dependence of the heat capacity on the isotopic mass has also been investigated recently for monatomic crystals such as diamond, silicon, and germanium. Multi-atomic crystals are expected to exhibit a different dependence of these heat capacities on the masses of each of the isotopes present. These effects have not been investigated in the past. We also present \textit{first-principles} calculations of the dependence of the heat capacities of GaN, as a canonical binary material, on each of the Ga and N masses. We show that they are indeed different, as expected from the fact that the Ga mass affects mainly the acoustic, that of N the optic phonons. It is hoped that these calculations will encourage experimental measurements of the dependence of the heat capacity on isotopic masses in binary and more complex semiconductors.Comment: 12 pages, 5 Figures, submitted to PR

Demazure crystals of generalized Verma modules and a flagged RSK correspondence

We prove that the Robinson-Schensted-Knuth correspondence is a \gl_{\infty}-crystal isomorphism between two realizations of the crystal graph of a generalized Verma module with respect to a maximal parabolic subalgebra of \gl_{\infty}. %This extends the previously known result that %the RSK correspondence is an isomorphism of bicrystals or double %crystals. A flagged version of the RSK correspondence is derived in a natural way by computing a Demazure crystal graph of a generalized Verma module. As an application, we discuss a relation between a Demazure crystal and plane partitions with a bounded condition

Dispersions of ellipsoidal particles in a nematic liquid crystal

Colloidal particles dispersed in a partially ordered medium, such as a liquid crystal (LC) phase, disturb its alignment and are subject to elastic forces. These forces are long-ranged, anisotropic and tunable through temperature or external fields, making them a valuable asset to control colloidal assembly. The latter is very sensitive to the particle geometry since it alters the interactions between the colloids. We here present a detailed numerical analysis of the energetics of elongated objects, namely prolate ellipsoids, immersed in a nematic host. The results, complemented with qualitative experiments, reveal novel LC configurations with peculiar topological properties around the ellipsoids, depending on their aspect ratio and the boundary conditions imposed on the nematic order parameter. The latter also determine the preferred orientation of ellipsoids in the nematic field, because of elastic torques, as well as the morphology of particles aggregates.Comment: 31 pages, 11 figure