Related papers: Description of $\mathsf{B}(\infty)$ through Kashiw…

The crystals for finite dimensional representations of sl(n+1) can be realized using Young tableaux. The infinity crystal on the other hand is naturally realized using multisegments, and there is a simple description of the embedding of…

A new categorical crystal structure for the quantum affine algebras is presented. We introduce the extended crystal $\widehat{B}_{\mathfrak{g}}(\infty)$ for an arbitrary quantum group, which is the product of infinite copies of the crystal…

We explain extremal weight crystals over affine Lie algebras of infinite rank using combinatorial models: a spinor model due to Kwon, and an infinite rank analogue of Kashiwara-Nakashima tableaux due to Lecouvey. In particular, we show that…

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…

Tristan Bozec gave a definition of generalized quantum groups that extends the usual definition of quantum groups to finite quivers with loops at vertices, and he introduced a theory of generalized crystals for this new family of Hopf…

In his work on crystal bases \cite{Kas}, Kashiwara introduced a certain degeneration of the quantized universal enveloping algebra of a semi-simple Lie algebra $\mathfrak g$, which he called a quantum boson algebra. In this paper, we…

Let $\lambda = \sum_{i \in I_{0}} m_{i} \varpi_{i}$, with $m_{i} \in \mathbb{Z}_{\ge 0}$ for $i \in I_{0}$, be a level-zero dominant integral weight for an affine Lie algebra $\mathfrak{g}$ over $\mathbb{Q}$, where the $\varpi_{i}$, $i \in…

Kashiwara and Saito have defined a crystal structure on the set of irreducible components of Lusztig's quiver varieties. This gives a geometric realization of the crystal graph of the lower half of the quantum group associated to a…

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…

A group grading on a semisimple Lie algebra over an algebraically closed field of characteristic zero is special if its identity component is zero; it is pure if at least one of its components, other than the identity component, contains a…

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 $B(\Lambda)$ be a level $\ell$ highest weight crystal of the quantum affine algebra $U_q(A_n^{(1)})$. We construct an explicit crystal isomorphism between the geometric realization $\gB(\Lambda)$ of the crystal $B(\Lambda)$ using quiver…

We use the crystal isomorphisms of the Fock space to describe two maps on partitions and multipartitions which naturally appear in the crystal basis theory for quantum groups in affine type A and in the representation theory of Hecke…

Let U be the quantum group associated to a Lie algebra g of rank n. The negative part U^- of U has a canonical basis B with favourable properties, introduced by Kashiwara and Lusztig. The approaches of Kashiwara and Lusztig lead to a set of…

We give a proof of the cyclicity conjecture of Akasaka-Kashiwara, for simply laced types, via quiver varieties. We get also an algebraic characterization of the standard modules.

Let $G$ be a connected reductive group over $\CC$ and let $G^{\vee}$ be the Langlands dual group. Crystals for $G^{\vee}$ were introduced by Kashiwara as certain ``combinatorial skeletons'' of finite-dimensional representations of…

We introduce coplactic raising and lowering operators $E'_i$, $F'_i$, $E_i$, and $F_i$ on shifted skew semistandard tableaux. We show that the primed operators and unprimed operators each independently form type A Kashiwara crystals (but…

We describe a set $\mathcal{R}^{\infty}$ consisting of tuples of integer sequences and provide certain explicit maps on it. We show that this defines a semiregular crystal for $\mathfrak{sl}_{n+1}$ and $\mathfrak{sp}_{2n}$ respectively.…

We give a construction of the compact real form of the Lie algebra of type $E_6$, using the finite irreducible subgroup of shape $3^{3+3}:\mathrm{SL}_3(3)$, which is isomorphic to a maximal subgroup of the orthogonal group $\Omega_7(3)$. In…

We prove a series of conjectures of Enomoto and Kashiwara on canonical bases and branching rules of affine Hecke algebras of type B. The main ingredient of the proof is a new graded Ext-algebra associated with quiver with involutions that…