Related papers: Kirillov-Reshetikhin crystals $B^{7,s}$ for type $…

Colloidal crystals formed by size-asymmetric binary particles co-assemble into a wide variety of colloidal compounds with lattices akin to ionic crystals. Recently, a transition from a compound phase with a sublattice of small particles to…

Building on ideas of Berthelot, we develop a crystalline cohomology formalism over divided power rings $(A, I_0, \eta)$ for any ring $A$, allowing $\mathbf{Z}$-flat $A$. For a smooth $A$-scheme $Y$ and a closed subscheme $X$ of $Y$ for…

The crystal structure of a disordered form of Cr$_{1/3}$NbS$_2$ has been characterized using diffraction and inelastic scattering of synchrotron radiation. In contrast to the previously reported symmetry (P6$_3$22), the crystal can be…

We construct atomic decompositions for crystals of type $C_{2}$ and define a charge statistic on them, thus providing positive combinatorial formulas for Kostka-Foulkes polynomials associated to them together with a natural geometric…

We describe a cluster algebra algorithm for calculating q-characters of Kirillov-Reshetikhin modules for any untwisted quantum affine algebra. This yields a geometric q-character formula for tensor products of Kirillov-Reshetikhin modules.…

Colored planar rook algebra is a semigroup algebra in which the basis element has a diagrammatic description. The category of finite dimensional modules over this algebra is completely reducible and suitable functors are defined on this…

Efficient heuristics have predicted many functional materials such as high-temperature superconducting hydrides, while inorganic structural chemistry explains why and how the crystal structures are stabilized. Here we develop the paired…

In this study, we present atomic models of nanoparticles exhibiting seventh-order rotational symmetry. We established that the point group symmetry of these objects corresponds to the dihedral group D7v. To gain a deeper understanding of…

What might a combinatorial interpretation of the Kronecker coefficients even look like? We introduce a class of combinatorial objects called bitableaux, which we believe are a natural candidate, and we formulate a purely combinatorial…

We study integral structures of crystalline representations over an unramified extension $K / \mathbb{Q}_p$ with the help of an auxillary ring $A_{\textrm{exp}}$. This ring has the nice property that it contains the the fundamental period…

The crystals for a finite-dimensional complex reductive Lie algebra $\mathfrak{g}$ encode the structure of its representations, yet can also reveal surprising new structure of their own. We study the cactus group $C_{\mathfrak{g}}$,…

We report the formation of a binary crystal of hard polyhedra due solely to entropic forces. Although the alternating arrangement of octahedra and tetrahedra is a known space-tessellation, it had not previously been observed in…

We give a crystal structure on the set of all irreducible components of Lagrangian subvarieties of quiver varieties. One con show that, as a crystal, it is isomorphic to the crystal base of an irreducible highest weight representation of a…

We review the differences in first order phase transition of single and multi-component systems, and then discuss the crystalline structure expected to exist in the mixed confined deconfined phase of hadronic matter. The particular context…

Fontaine's $D_{\mathrm{cris}}$ functor allow us to associate an isocrystal to any crystalline representation. For a reductive group $G$, we study the reduction of lattices inside a germ of crystalline representations with $G$-structure $V$…

We develop a theory of bicrystalline ideals, synthesizing Gr\"obner basis techniques and Kashiwara's crystal theory. This provides a unified algebraic, combinatorial, and computational approach that applies to ideals of interest, old and…

For a Kac-Moody group $G$, double Bruhat cells $G^{u,e}$ ($u$ is a Weyl group element) have positive geometric crystal structures. In arXiv:1210.2533, it is shown that there exist birational maps between `cluster tori'…

Under the exponential parametrization scheme $ m^{\nu}_{ij}\sim r e^{i \theta}$, we propose a \emph{unique} neutrino mass matrix texture that exhibits four correlations among its elements. The mixing scheme obtained from the proposed…

Several combinatorial problems of (quasi-)crystallography are reviewed with special emphasis on a unified approach, valid for both crystals and quasicrystals. In particular, we consider planar sublattices, similarity sublattices,…

We investigate the interplay of crystal bases and completions in the sense of Enright on certain nonintegrable representations of quantum groups. We define completions of crystal bases, show that this notion of completion is compatible with…