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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…
We give (conjectural) analogs of Berenstein-Zelevinsky data for affine type $A$. Moreover, by using these affine analogs of Berenstein-Zelevinsky data, we realize the crystal basis of the negative part of the quantized universal enveloping…
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.…
We classify the elements of $W(\tilde{A}_n)$ by giving a canonical reduced expression for each, using basic tools among which affine length. We give some direct consequences for such a canonical form: a description of left multiplication by…
We consider the standard modules of rectangular highest weights of affine Lie algebras in types $A_{2l-1}^{(2)}$ and $D_{l+1}^{(2)}$. By using vertex algebraic techniques we construct the combinatorial bases for standard modules and their…
Using the bases of principal subspaces for twisted affine Lie algebras except $A_{2l}^{(2)}$ by Butorac and Sadowski, we construct bases of the highest weight modules of highest weight $k\Lambda_0$ and parafermionic spases for the same…
Elpasolite is the predominant quaternary crystal structure (AlNaK$_2$F$_6$ prototype) reported in the Inorganic Crystal Structure Database. We have developed a machine learning model to calculate density functional theory quality formation…
In this paper, we study a tensor product of perfect Kirillov-Reshetikhin crystals (KR crystals for short) whose levels are not necessarily equal. We show that, by tensoring with a certain highest weight element, such a crystal becomes…
This paper is concerned with one-dimensional sums in classical affine types. We prove a conjecture of the third author and Zabrocki by showing they all decompose in terms of one-dimensional sums related to affine type A provided the rank of…
In this paper we study the geometric structure of affine Deligne-Lusztig varieties for $GL_n$ and $b$ basic. We introduce a new condition on $\lambda$. If this is satisfied, then the corresponding affine Deligne-Lusztig variety is the…
Using Crystal basis theory, we study the relative Gel'fand-Kalinin-Fuks cohomology groups of the formal Hamiltonian vector fields on 6-dimensional plane with weight =2,4,6.
We present a combinatorial model, called \emph{perforated tableaux}, to study $A_{n-1}$ crystals, unifying several previously studied combinatorial models. We identify nodes in the $k$-fold tensor product of the standard crystal with length…
We construct every finite-dimensional irreducible representation of the simple Lie algebra of type $\mathsf{E}_{7}$ whose highest weight is a nonnegative integer multiple of the dominant minuscule weight associated with the type…
Quasicrystals described as the projections of higher dimensional cubic lattices, and the particular affine extensions of the dihedral group $I_2(h)$ of order $2h$, $h=2n$ being the Coxeter number, as a subgroup of affine $B_n$ offers a…
For every non-exceptional affine Lie algebra, we explicitly construct a positive geometric crystal associated with a fundamental representation. We also show that its ultra-discretization is isomorphic to the limit of certain perfect…
We are interested in the structure of the crystal graph of level $l$ Fock spaces representations of $\mathcal{U}_q (\widehat{\mathfrak{sl}_e})$. Since the work of Shan [26], we know that this graph encodes the modular branching rule for a…
We derive decomposition formulas for supercharacters of quantum affine ortho-symplectic superalgebras and twisted quantum affine superalgebras into supercharacters of their finite-type quantum sub-superalgebras, by employing Cauchy-type…
We propose a method to prove a polyhedral branching formula for Kirillov-Reshetikhin (KR) modules over a quantum affine algebra. When the underlying simple Lie algebra is of exceptional type, such a formula remains conjectural in many…
F-theory is a non-perturbative formulation of type IIB superstring theory which allows for the decoupling of gravity and for the formulation of GUT theories based on the gauge group E6. In this paper we explore F-theory models in which the…
In this note, we compute the group of automorphisms of Projective, Affine and Euclidean Geometries in the sense of Klein. As an application, we give a simple construction of the outer automorphism of S_6.