Related papers: $B_2$-crystals: axioms, structure, models
In 2012, we discovered superconductivity in Bi4O4S3 and LaO1-xFxBiS2, whose crystal structures are composed of alternate stacks of a BiS2-type conduction layer and a blocking layer. Since the discovery, many related superconductors and…
First we develop the theory of local rules for coboundary categories. Then we describe the local rules in two main cases. First for the quantum groups in general and in the seminormal representations of the Hecke algebras. Then for crystals…
New ultra-hard rhombohedral B2N2 and BC2N - or hexagonal B6N6 and B3C6N3 - are derived from 3R graphite based on crystal chemistry rationale schematizing a mechanism for 2D => 3D transformation. Full unconstrained geometry optimizations…
Let $V(\lambda)$ be the irreducible lowest weight $U_q(D(N,1))$-module with lowest weight $\lambda$. Assume $\lambda = n_0\omega_0-\sum_{i=0}^{N}n_i\omega_i$, where $\omega_0$ is the fundamental weight corresponding to the unique odd coroot…
The vertices of any (combinatorial) Kashiwara crystal graph carry a natural monoid structure given by identifying words labelling vertices that appear in the same position of isomorphic components of the crystal. Working on a purely…
In the paper we prove that every automorphism of any adjoint Chevalley group of types B_2 or G_2 is standard, i.e., it is a composition of the ``inner'' automorphism, ring automorphism and central automorphism.
In this paper we study local cohomology of finitely generated bigraded modules over a standard bigraded ring with respect to the irrelevant bigraded ideals and establish a duality theorem. Several applications are considered.
The rigged configuration realization $RC(\infty)$ of the crystal $B(\infty)$ was originally presented as a certain connected component within a larger crystal. In this work, we make the realization more concrete by identifying the elements…
Consider Kashiwara's crystal associated to a highest weight representation of a symmetric Kac-Moody algebra. There is a geometric realization of this object using Nakajima's quiver varieties, but in many particular cases it can also be…
A generic method for combinatorial constructions of intrinsic geometrical spaces is presented. It is based on the well known inverse sequences of finite graphs that determine (in the limit) topological spaces. If a pattern of the…
We investigate structural and combinatorial properties of Bi-Cayley graphs defined over cyclic groups of order $p^2q^2$, where $p$ and $q$ are distinct primes. We begin by describing their fundamental group-theoretic underpinnings. The main…
This continues the investigation of a combinatorial model for the variation of dynamics in the family of rational maps of degree two, by concentrating on those varieties in which one critical point is periodic. We prove some general results…
A new graph-based order parameter is introduced for the characterization of atomistic structures. The order parameter is universal to any material/chemical system, and is transferable to all structural geometries. Three sets of data are…
We show that many tame modules of the quantum toroidal $\mathfrak{gl}_2$ algebra can be explicitly constructed in a purely combinatorial way using the theory of $q$-characters. The examples include families of evaluation modules obtained…
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…
This is a book on higher-categorical diagrams, including pasting diagrams. It aims to provide a thorough and modern reference on the subject, collecting, revisiting and expanding results scattered across the literature, informed by recent…
Solid solutions of SiO2 and B2O3 in Li2O 2SiO2 are synthesized and characterized for the first time. Their structure and crystallization mechanisms are investigated employing a combination of simulations at the density functional theory…
Novel ordered intermetallic compounds have stimulated much interest. Ru-Al alloys are a prominent class of high-temperature structural materials, but the experimentally reported crystal structure of the intermetallic Ru2Al5 phase remains…
Crystal structures are defined by the periodic arrangement of atoms in 3D space, inherently making them equivariant to SO(3) group. A fundamental requirement for crystal property prediction is that the model's output should remain invariant…
We define three combinatorial models for \hat{sl(n)} crystals, parametrized by partitions, configurations of beads on an `abacus', and cylindric plane partitions, respectively. These are reducible, but we can identify an irreducible…