Related papers: Crystal constructions in Number Theory

Weyl group multiple Dirichlet series are Dirichlet series in $r$ complex variables, with analytic continuation to $\mathbb{C}^r$ and a group of functional equations isomorphic to the Weyl group of a reduced root system of rank $r$. Such…

By studying the structures of clusters bound by a model potential that favours polytetrahedral order, we find a previously unknown series of `magic numbers' (i.e. sizes of special stability) whose polytetrahedral structures are…

All algebras in a very large, axiomatically defined class of quantum nilpotent algebras are proved to possess quantum cluster algebra structures under mild conditions. Furthermore, it is shown that these quantum cluster algebras always…

We consider decompositions of digraphs into edge-disjoint paths and describe their connection with the $n$-th Weyl algebra of differential operators. This approach gives a graph-theoretic combinatorial view of the normal ordering problem…

A Littelmann path model is constructed for crystals pertaining to a not necessarily symmetrizable Borcherds-Cartan matrix. Here one must overcome several combinatorial problems coming from the imaginary simple roots. The main results are an…

Mekler's construction is a powerful technique for building purely algebraic structures from combinatorial ones. Its power lies in the fact that it allows various model-theoretic tameness properties of the combinatorial structure to transfer…

We determine the blocks of the walled Brauer algebra in characteristic zero. These can be described in terms of orbits of the action of a Weyl group of type $A$ on a certain set of weights. In positive characteristic we give a linkage…

The manuscript reviews Dirichlet Series of important multiplicative arithmetic functions. The aim is to represent these as products and ratios of Riemann zeta-functions, or, if that concise format is not found, to provide the leading…

Crystal structures can be viewed as assemblies of space-filling polyhedra, which play a critical role in determining material properties such as ionic conductivity and dielectric constant. However, most conventional crystal structure…

Equilibrium polyethylene crystal structure, cohesive energy, and elastic constants are calculated by density-functional theory applied with a recently proposed density functional (vdW-DF) for general geometries [Phys. Rev. Lett. 92, 246401…

We propose a relatively new notion of two-valued elements, which arises naturally in constructing the star exponential functions of the quad-ratics in the Weyl algebra over the complex number field. This notion enables us to describe the…

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…

Functions whose symmetries form a crystallographic group in particular have a lattice of periods, and the set of their level curves forms a periodic pattern. We show how after projecting these functions, one obtains new functions with a…

We introduce certain directed multigraphs with extra structure, called Weyl graphs, which model quotients of Tits buildings by type-preserving chamber-free group actions. Their advantage over complexes of groups, which are often used for…

Regular $A_n$-crystals are certain edge-colored directed graphs which are related to representations of the quantized universal enveloping algebra $U_q(\mathfrak{sl}_{n+1})$. For such a crystal $K$ with colors $1,2,...,n$, we consider its…

Molecular and polymeric crystals show a wide range of functional properties that arise from the interplay between the atomic-scale structure of their constituent molecules and the organization of these molecules within the crystal lattice…

An important problem from invariant theory is to describe the subspace of a tensor power of a representation invariant under the action of the group. According to Weyl's classic, the first main (later: 'fundamental') theorem of invariant…

Let $\mathfrak{g}$ be a Lie algebra all of whose regular subalgebras of rank 2 are type $A_{1}\times A_{1}$, $A_{2}$, or $C_{2}$, and let $B$ be a crystal graph corresponding to a representation of $\mathfrak{g}$. We explicitly describe the…

We define a multiple Dirichlet series whose group of functional equations is the Weyl group of the affine Kac-Moody root system $\tilde{A}_n$, generalizing the theory of multiple Dirichlet series for finite Weyl groups. The construction is…

We prove functional equations for multiple Dirichlet series defined by a collection of five geometric axioms. We find functional equations of two types: one modeled on the functional equations of Dirichlet $L$-functions, and another modeled…