Related papers: Crystal structure on rigged configurations and the…
The Ricci form is a moment map for the action of the group of exact volume preserving diffeomorphisms on the space of almost complex structures. This observation yields a new approach to the Weil-Petersson symplectic form on the Teichmuller…
In this note, inspired by the proof of the Kirillov-Reshetikhin conjecture, we consider tensor products of Kirillov-Reshetikhin modules of a fixed node and various level. We fix a positive integer and attach to each of its partitions such a…
Autonomous materials discovery with desired properties is one of the ultimate goals for materials science, and the current studies have been focusing mostly on high-throughput screening based on density functional theory calculations and…
Level-restricted paths play an important role in crystal theory. They correspond to certain highest weight vectors of modules of quantum affine algebras. We show that the recently established bijection between Littlewood--Richardson…
Kashiwara and Saito have a geometric construction of the infinity crystal for any symmetric Kac-Moody algebra. The underlying set consists of the irreducible components of Lusztig's quiver varieties, which are varieties of nilpotent…
The design of crystal materials plays a critical role in areas such as new energy development, biomedical engineering, and semiconductors. Recent advances in data-driven methods have enabled the generation of diverse crystal structures.…
In this paper, we study the affine Springer fiber $\mathcal{F} l_N$ in type $A$ for rectangular type semisimple nil-element $N$ and calculate the relative position between irreducible components. In particular, we use the affine matrix ball…
We define and study the Kirillov--Reshetikhin modules for algebras of type $G_2$. We compute the graded character of these modules and verify that they are in accordance with the conjectures in math.QA/981202 and math.QA/0102113. These…
A new $n-$ noded polygonal plate element is proposed for the analysis of plate structures comprising of thin and thick members. The formulation is based on the discrete Kirchhoff Mindlin theory. On each side of the polygonal element,…
Regular $A_n$-, $B_n$- and $C_n$-crystals are edge-colored directed graphs, with ordered colors $1,2,...,n$, which are related to representations of quantized algebras $U_q(\mathfrak{sl}_{n+1})$, $U_q(\mathfrak{sp}_{2n})$ and…
We propose a mathematical description of crystal structure: underlying translational periodicity together with the distinct atomic positions up to the symmetry operations in the unit cell. It is consistent with the international table of…
The optimal "twisted" geometry of a crystalline layer on a crystal is long known, but that on a quasicrystal is still unknown and open. We predict analytically that the layer equilibrium configuration will generally exhibit a nonzero misfit…
In [Frieden, arXiv:1706.02844], we constructed a geometric crystal on the variety $\mathbb{X}_{k} := {\rm Gr}(k,n) \times \mathbb{C}^\times$ which tropicalizes to the affine crystal structure on rectangular tableaux with $n-k$ rows. In this…
Lusztig $q$-weight multiplicities extend the Kostka-Foulkes polynomials to a broader range of Lie types. In this work, we investigate these multiplicities through the framework of Kirillov-Reshetikhin crystals. Specifically, for type $C$…
Materials discovery via high-throughput methods relies on the availability of structural prototypes, which are generally decorated with varying combinations of elements to produce potential new materials. To facilitate the automatic…
In this paper we introduce geometric crystals and unipotent crystals which are algebro-geometric analogues of Kashiwara's crystal bases. Given a reductive group G, let I be the set of vertices of the Dynkin diagram of G and T be the maximal…
Elucidating the interplay of stress and geometry is a fundamental scientific question arising in multiple fields. In this work, we investigate the geometric frustration of crystalline caps confined on the sphere in both elastic and plastic…
We introduce a computational method to optimize target physical properties in the full configuration space regarding atomic composition, chemical stoichiometry, and crystal structure. The approach combines the universal potential of the…
Given recipe of qualitative, kinetic modelling by geometric methods of three-dimensional dendritic crystals. Characteristic features of the perturbations appearing on the surface of a spherical body, leading to different scenarios of the…
We establish the equality of the specialization $P_\lambda(x;q,0)$ of the Macdonald polynomial at $t=0$ with the graded character $X_\lambda(x;q)$ of a tensor product of "single-column" Kirillov-Reshetikhin (KR) modules for untwisted affine…