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We study graded dimension formulas for finite quiver Hecke algebras $R^{\Lambda_0}(\beta)$ of type $A^{(2)}_{2\ell}$ and $D^{(2)}_{\ell+1}$ using combinatorics of Young walls. We introduce the notion of standard tableaux for proper Young…
We develop a combinatorial model of networks on orientable surfaces, and study weight and homology generating functions of paths and cycles in these networks. Network transformations preserving these generating functions are investigated.…
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.…
We consider the affine Lie algebra $\widehat{\mathfrak{sl}_2}$ and the Kostant-Kumar submodules of tensor products of its level 1 highest weight integrable representations. We construct crystals for these submodules in terms of the charged…
There are two parts to this work, which are largely independent. The first consists of a series of results concerning the crystal commutor of Henriques and Kamnitzer. We first describe the relationship between the crystal commutor and…
The Ram-Yip formula for Macdonald polynomials (at t=0) provides a statistic which we call charge. In types A and C it can be defined on tensor products of Kashiwara-Nakashima single column crystals. In this paper we prove that the charge is…
The effective interaction between two planar walls immersed in a fluid is investigated by use of Density Functional Theory in the super-critical region of the phase diagram. A hard core Yukawa model of fluid is studied with special…
In this paper the relationship between the density functional theory of freezing and phase field modeling is examined. More specifically a connection is made between the correlation functions that enter density functional theory and the…
In this work, an expression for the affine weight is calculated for Nakajima monomial crystals in affine types $A_n^{(1)}$ and $B_3^{(1)}$.
We prove that, in types $E_{6,7,8}^{(1)}$, $F_4^{(1)}$ and $E_6^{(2)}$, every Kirillov--Reshetikhin module associated with the node adjacent to the adjoint one (near adjoint node) has a crystal pseudobase, by applying the criterion…
In this paper we report metamaterial properties including negative and singular effective properties for what would traditionally be considered non locally resonant 2-phase phononic unit cells. The negative effective material properties…
We consider a category of $\gl_\infty$-crystals, whose objects are disjoint unions of extremal weight crystals of non-negative level with certain finite conditions on the multiplicity of connected components. We show that it is a monoidal…
As shown by Stembridge, crystal graphs can be characterized by their local behavior. In this paper, we observe that a certain local property on highest weight crystals forces a more global property. In type $A$, this statement says that if…
Different computational techniques in combination with molecular dynamics computer simulation are used to to determine the wall-liquid and the wall-crystal interfacial free energies of a modified Lennard-Jones (LJ) system in contact with a…
First-principles calculations within the DFT are employed to investigate the relationship between the electronic structure and the unexpected features of the hexagonal cell parameters of YbAuIn and Yb$_{3}$AuGe$_{2}$In$_{3}$. Calculations…
We characterize, in the case of affine sl(2), the crystal base of the Demazure module E_w(\La) in terms of extended Young diagrams or paths for any dominant integral weight \La and Weyl group element w. Its character is evaluated via two…
The formation of heterogeneous lanthanide-ion clusters in CaF$_2$ was investigated experimentally and computationally. CaF$_2$ nanoparticles co-doped with 20~mol\% Yb$^{3+}$ and 2~mol\% Ln$^{3+}$ (Ln$^{3+}$ = Ce$^{3+}$, Pr$^{3+}$,…
We derive new crystal melting models from Chern-Simons theory on the three-sphere. Via large N duality, these models compute amplitudes for A-model on the resolved conifold. The crystal is bounded by two walls whose distance corresponds to…
A relationship is established between the effective Youngs modulus of a two-phase elastic composite and a known mathematical mean value. Specifically, the effective Youngs modulus of a composite obtained from repeated parallel and serial…
We construct by fusion product new irreducible representations of the quantum affinization $U_q(\hat{sl}_\infty)$. The action is defined via the Drinfeld coproduct and is related to the crystal structure of semi-standard tableaux of type…