Related papers: Crystal bases as tuples of integer sequences

Let ${\cal M}(S; \Lambda; P)$ denote a Rees $I\times \Lambda$ matrix semigroup without zero over a semigroup $S$, where $I$ is a singleton. If $\theta _S$ denotes the kernel of the right regular representation of a semigroup $S$, then a…

We define the Higgs algebra $\mathcal{H}_\P1$ of the projective line, as a convolution algebra of constructible functions on the global nilpotent cone $\underline{\Lambda}_\P1$, a lagrangian substack of the Higgs bundle $T^*\Coh_\P1$, where…

A ring $R$ is called right SSP (SIP) if the sum (intersection) of any two direct summands of $R_{R}$ is also a direct summand. Left sides can be defined similarly. The following are equivalent: (1) $R$ is right SSP. (2) $R$ is right C3 and…

Let $U_q(\frak{g})$ a the quantum affine algebra of type $A_n^{(1)}$, $A_{2n-1}^{(2)}$, $A_{2n}^{(2)}$, $B_n^{(1)}$, $D_n^{(1)}$ and $D_{n+1}^{(2)}$, and let $\mathcal{F}(\Lambda)$ be the Fock space representation for a level 1 dominant…

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…

Let B be the crystal basis of the minus part of the quantized enveloping algebra of a semi-simple Lie algebra. Kashiwara has shown that B has a combinatorial description in terms of an embedding of B into the tensor product of B and k…

Let $\mathrm{d}(A)$ be the asymptotic density (if it exists) of a sequence of integers $A$. For any real numbers $0\leq\alpha\leq\beta\leq 1$, we solve the question of the existence of a sequence $A$ of positive integers such that…

By utilizing the combinatorial properties of various tableau models, we establish an explicit correspondence between the polyhedral realizations of the crystal bases $\mathcal B(\lambda)$ (resp. $\mathcal B(\infty)$) of type $A_n$ and the…

We study the semigroup extension $\mathscr{I}_\lambda^n(S)$ of a semigroup $S$ by symmetric inverse semigroups of a bounded finite rank. We describe idempotents and regular elements of the semigroups $\mathscr{I}_\lambda^n(S)$ and…

A ring R is said to be VNL if for any a in R, either a or 1-a is (von Neumann) regular. The class of VNL rings lies properly between the exchange rings and (von Neumann) regular rings. We characterize abelian VNL rings. We also characterize…

We introduce several commutative rings, the snake rings, that have strong connections to cluster algebras. The elements of these rings are residue classes of unions of certain labeled graphs that were used to construct canonical bases in…

We consider standard graded toric rings $R_{\Delta}$ whose generators correspond to the faces of a simplicial complex $\Delta$. When $R_{\Delta}$ is normal, it is shown that its divisor class group is free. For a flag complex $\Delta$ which…

For each $n\geqslant3$, we construct an uncountable family of models of the crystal of the basic $U_q(\hat{\mathfrak{sl}}_n)$-module. These models are all based on partitions, and include the usual $n$-regular and $n$-restricted models, as…

We give a general way of representing the crystal (base) corresponding to the intgrable highest weight modules of quantum Kac-Moody algebras, which is called polyhedral realizations. This is applied to describe explicitly the crystal bases…

Fix a simply-laced semisimple Lie algebra. We study the crystal $ B(n\lambda)$, were $\lambda$ is a dominant minuscule weight and $n$ is a natural number. On one hand, $B(n\lambda)$ can be realized combinatorially by height $n$ reverse…

For a nonempty subset $X$ of a ring $R$, the ring $R$ is called $X$-semiprime if, given $a\in R$, $aXa=0$ implies $a=0$. This provides a proper class of semiprime rings. First, we clarify the relationship between idempotent semiprime and…

We use Khovanov-Lauda-Rouquier (KLR) algebras to categorify a crystal isomorphism between a fundamental crystal and the tensor product of a Kirillov-Reshetikhin crystal and another fundamental crystal, all in affine type. The nodes of the…

The crystal bases are quite useful combinatorial tools to study the representations of quantized universal enveloping algebras $U_q(\mathfrak{g})$. The polyhedral realization for $B(\infty)$ is a combinatorial description of the crystal…

Let $S$ be a semigroup, $\Lambda$ a non-empty set and $P$ a mapping of $\Lambda$ into $S$. The set $S\times \Lambda$ together with the operation $\circ _P$ defined by $(s, \lambda)\circ _P(t, \mu )=(sP(\lambda)t, \mu )$ form a semigroup…

We construct a crystal base of $U_q(\mathfrak{gl}(m|n))^-$, the negative half of the quantum superalgebra $U_q(\mathfrak{gl}(m|n))$. We give a combinatorial description of the associated crystal $\mathscr{B}_{m|n}(\infty)$, which is equal…