Related papers: Crystals from 5-vertex ice models
I examine a variety of snow crystal growth measurements taken at a temperature of -5 C, as a function of supersaturation, background gas pressure, and crystal morphology. Both plate-like and columnar prismatic forms are observed under…
We develop a model for the reflection and transmission of plane waves by an isotropic layer sandwiched between two uniaxial crystals of arbitrary orientation. In the laboratory frame, reflection and transmission coefficients corresponding…
We consider nematic liquid crystals in a bounded, convex polyhedron described by a director field n(r) subject to tangent boundary conditions. We derive lower bounds for the one-constant elastic energy in terms of topological invariants.…
Let B be the Lie algebra with basis {L_{i,j},C|i,j\in Z} and relations [L_{i,j},L_{k,l}]=((j+1)k-i(l+1))L_{i+k,j+l}+i\delta_{i,-k}\delta_{j+l,-2}C, [C,L_{i,j}]=0. It is proved that an irreducible highest weight B-module is quasifinite if…
In this short note we announce three formulas for the set of weights of various classes of highest weight modules $\V$ with highest weight \lambda, over a complex semisimple Lie algebra $\lie{g}$ with Cartan subalgebra $\lie{h}$. These…
Let $n\ge 2$ be a positive integer. To each irreducible representation $\sigma$ of $\mathrm{Sp}(1)$, an $\mathrm{Sp}(1)$-Kepler problem in dimension $(4n-3)$ is constructed and analyzed. This system is super integrable and when $n=2$ it is…
For $n \geq 2$ consider the affine Lie algebra $\widehat{s\ell}(n)$ with simple roots $\{\alpha_i \mid 0 \leq i \leq n-1\}$. Let $V(k\Lambda_0), \, k \in \mathbb{Z}_{\geq 1}$ denote the integrable highest weight $\widehat{s\ell}(n)$-module…
Recent experiments have created supersolids of dipolar quantum droplets. The resulting crystals lack, however, a genuine cohesive energy and are maintained by the presence of an external confinement, bearing a resemblance to the case of ion…
Let $\mathfrak{g}$ be a finite-dimensional simple complex Lie algebra. A layer sum is introduced as the sum of formal exponentials of the distinct weights appearing in an irreducible $\mathfrak{g}$-module. It is argued that the character of…
Berthelot's conjecture predicts that under a proper and smooth morphism of schemes in characteristic $p$, the higher direct images of an overconvergent $F$-isocrystal are overconvergent $F$-isocrystals. In this paper we prove that this is…
Crystal base of the level 0 part of the modified quantum affine algebra $\widetilde U_q(\widehat{sl_2})_0$ is given by path. Weyl group actions, extremal vectors and crystal structure of all irreducible components are described explicitly.
The interface of a cholesteric liquid crystal with an isotropic fluid can display a range of unusual properties, such as a layer of topological defects close to an undulated interface. These properties have been know for a long time and…
In earlier work with C.~Monical, we introduced the notion of a K-crystal, with applications to K-theoretic Schubert calculus and the study of Lascoux polynomials. We conjectured that such a K-crystal structure existed on the set of…
The result that near the melting point three-dimensional crystals have an octahedronic structure is generalized to higher flat non compact dimensions.
The number of distinguishable inherent structures of a liquid is the key component to understanding the thermodynamics of glass formers. In the case of hard potential systems such as hard discs, spheres and ellipsoids, an inherent structure…
In the first part of the paper we give the denominator identity for all simple finite-dimensional Lie super algebras $\frak g\/$ with a non-degenerate invariant bilinear form. We give also a character and (super) dimension formulas for all…
In this continuation of \cite{BM}, we prove the following: Let $\Gamma\subset \text{SL}(2,{\mathbb C})$ be a cocompact lattice, and let $\rho: \Gamma \rightarrow \text{GL}(r,{\mathbb C})$ be an irreducible representation. Then the…
To find crystals of $\mathfrak{sl}_2$ representations of the form $\Lambda^n\text{Sym}^r\mathbb{C}^2$ it suffices to solve the combinatorial problem of decomposing Young's lattice into symmetric, saturated chains. We review the literature…
We will show that the complex shapes of snow crystals can be explained from a simple basic mechanism that is also responsible for the appearance of many others structures in nature. We expect that this new physical mechanism, that follows…
We demonstrate that higher-order electric susceptibilities in crystals can be enhanced and understood through nontrivial topological invariants and quantum geometry, using one-dimensional $\pi$-conjugated chains as representative model…