Related papers: A new Young wall realization of $B(\lambda)$ and $…

Fix any Borcherds-Kac-Moody $\mathbb{C}$-Lie algebra (BKM LA) $\mathfrak{g}=\mathfrak{g}(A)$ of BKM-Cartan matrix $A$, and Cartan subalgebra $\mathfrak{h}\subset \mathfrak{g}$. In this paper, we obtain explicit weight formulas of any…

For a simply connected connected simple algebraic group $G$, it is known that a variety $B_{w_0}^-:=B^-\cap U\overline{w_0}U$ has a geometric crystal structure with a positive structure…

We classify the simple infinite dimensional integrable modules with finite dimensional weight spaces over the quantized enveloping algebra of an untwisted affine algebra. We prove that these are either highest (lowest) weight integrable…

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 combinatorial R matrices are obtained for a family {B_l} of crystals for U'_q(C^{(1)}_n) and U'_q(A^{(2)}_{2n-1}), where B_l is the crystal of the irreducible module corresponding to the one-row Young diagram of length l. The…

Let $\mu$ be a non-negative Borel measure on $R^d$ satisfying that the measure of a cube in $R^d$ is smaller than the length of its side raised to the $n$-th power, $0<n\leq d$. In this article we study the class of weights related to the…

Let $g$ be an affine Lie algebra with index set $I = \{0, 1, 2, \cdots , n\}$ and $g^L$ be its Langlands dual. It is conjectured that for each Dynkin node $i \in I \setminus \{0\}$ the affine Lie algebra $g$ has a positive geometric crystal…

Let $V(\Lambda_i)$ (resp., $V(-\Lambda_j)$) be a fundamental integrable highest (resp., lowest) weight module of $U_q(\hat{sl}_{2})$. The tensor product $V(\Lambda_i)\otimes V(-\Lambda_j)$ is filtered by submodules…

In the recent papers with Masaki Kashiwara, the author introduced the notion of symmetric crystals and presented the Lascoux-Leclerc-Thibon-Ariki type conjectures for the affine Hecke algebras of type $B$. Namely, we conjectured that…

We obtain a graded character formula for certain graded modules for the current algebra over a simple Lie algebra of type E6. For certain values of their highest weight, these modules were conjectured to be isomorphic to the classical limit…

It is shown that the direct limit of the semistandard decomposition tableau model for polynomial representations of the queer Lie superalgebra exists, which is believed to be the crystal for the upper half of the corresponding quantum…

We construct an explicit algorithm of the static-preserving bijection between the rigged configurations and the highest weight paths of the form $(B^{2,1})^{\otimes L}$ in the $G_{2}^{(1)}$ adjoint crystals.

We develop a new method for estimating the decay probability of the false vacuum via regularized instantons. Namely, we consider the case where the potential is either unbounded from below or the second minimum corresponding to the true…

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 give a combinatorial construction, not involving a presentation, of almost all untwisted affine Kac--Moody algebras modulo their one-dimensional centres in terms of signed raising and lowering operators on a certain distributive lattice…

We realize (via an explicit isomorphism) the walled Brauer algebra for an arbitrary integral parameter delta as an idempotent truncation of a level two cyclotomic degenerate affine walled Brauer algebra. The latter arises naturally in Lie…

Motivated by the work of Nakayashiki on the inhomogeneous vertex models of 6-vertex type, we introduce the notion of crystals with head. We show that the tensor product of the highest weight crystal of level k and the perfect crystal of…

We study the monomial crystal defined by the second author. We show that each component of the monomial crystal can be embedded into a crystal of an extremal weight module introduced by Kashiwara. And we determine all monomials appearing in…

In this paper, we study level-zero extremal weight modules over twisted quantum affine algebras. To this end, we introduce semi-infinite Lakshmibai--Seshadri paths associated with a level-zero dominant integral weight $\lambda$. We then…

We establish a degeneration isomorphism between quantum toroidal algebras and untwisted affine Yangians, valid for all untwisted affine Kac-Moody Lie algebras. Specifically, we prove that the affine Yangian $Y_\hbar(\mathfrak{g})$ is…