Highest Weight Crystals Research Articles

Zigzag strip bundles and crystals II

Zigzag strip bundles are combinatorial models realizing the crystals B(∞) and the highest weight crystals B(λ) over quantum affine algebras Uq(g), and they are closely related with Nakajima monomials and Kashiwara embeddings of specific type. In this paper, we introduce new zigzag strip bundles for Uq(g), which we call reverse zigzag strip bundles in order to distinguish from the previous ones, and we give explicit 1-1 correspondences with the set of Nakajima monomials and the image of Kashiwara embedding which are different from those appeared in the study of the zigzag strip bundles. Another main result is to give different points of view about zigzag strip bundles and reverse zigzag strip bundles. Indeed, the previous 1-1 correspondences between the sets of zigzag strip bundles or reverse zigzag strip bundles, and the sets of Nakajima monomials or images of Kashiwara embeddings were determined based on the number of blocks in the columns of these strip bundles. In our work, we use the number of blocks in the rows of zigzag strip bundles and reverse zigzag strip bundles to provide characterizations of Nakajima monomials and images of Kashiwara embeddings that differ from the previous ones. That is, we give characterizations of Nakajima monomials and Kashiwara embeddings of 4 types. Further, we discuss the connection between zigzag strip bundles and reverse zigzag strip bundles.

Strip bundle realization of the crystals over Uq(D4(3))

We introduce new combinatorial models called strip bundles for the quantum affine algebra Uq(D4(3)) and give new realizations of the crystal B(∞) and the highest weight crystals B(λ) over Uq(D4(3)) in terms of strip bundles. In these strip bundle realizations, the highest weight crystal B(Λ1) of the natural representation V(Λ1) over the quantum finite algebra Uq(G2) plays a crucial role.

Strip bundle realization of the crystals over Uq(G2(1))

Motivated by the zigzag strip bundles which are combinatorial models realizing the crystals B(∞) for the quantum affine algebras Uq(g), where g=Bn(1),Dn(1),Dn+1(2),Cn(1), A2n−1(2),A2n(2), we introduce a new combinatorial model called strip bundles for the quantum affine algebra Uq(G2(1)). We give new realizations S(∞) and S(λ) of the crystal B(∞) and the highest weight crystals B(λ) over Uq(G2(1)) using strip bundles, and as subsets of S(∞) and S(λ), we also give realizations of the crystal B(∞) and the highest weight crystals B(λ) over the quantum finite algebra Uq(G2). Moreover, we give characterizations of the image of the crystal embedding Ψi and the connected component C1 in the set M of all Nakajima monomials which are isomorphic to the crystal B(∞) over Uq(G2(1)).

Strip Bundles and the Highest Weight Crystals Over Quantum Infinite Rank Affine Algebras

We give new realizations of the highest weight crystals B(λ) over the quantum infinite rank affine algebras U q (A ∞), U q (B ∞), U q (C ∞), and U q (D ∞) using strip bundles, which are related with Nakajima monomials.

Zigzag strip bundles and highest weight crystals

Zigzag strip bundles are new combinatorial models realizing the crystals B(∞) for the quantum affine algebras Uq(g), where g=Bn(1),Dn(1),Dn+1(2),Cn(1),A2n−1(2),A2n(2). In this paper, we give new realizations of the crystal bases B(λ) for the irreducible highest weight modules V(λ) over quantum affine algebras Uq(g) using zigzag strip bundles. Further, we discuss the connection between zigzag strip bundle realization, Nakajima monomial realization, and polyhedral realization of the crystals B(λ).

Crystal ℬ(∞) for Quantum Affine Algebras: A Young Wall Description

We study the crystal base ℬ(∞) associated with the negative part of the quantum group for classical affine Lie algebra types. Using the expression of the crystal ℬ(∞) as a union of highest weight crystals, we give an explicit description of ℬ(∞) in terms of Young walls. The new description realizes ℬ(∞) as a set of certain objects called marginally large walls.

Monomial Realization of the Tensor Product of Crystals for Quantum Finite Algebras

In this paper, we give a combinatorial description of the crystal bases of the tensor products of the irreducible highest weight modules over quantum finite algebras in terms of Nakajima monomials. Also, we explain how to decompose the tensor product of irreducible highest weight crystals into irreducibles by giving a bijection defined by multiplication of monomials.

Quiver Varieties and Adjoint Crystals of Level ℓ for Type A n (1)

Let B(Λ) be a level ℓ highest weight crystal of the quantum affine algebra . We construct an explicit crystal isomorphism between the geometric realization 𝔅(Λ) of the crystal B(Λ) using quiver varieties and the path realization 𝒫ad(Λ) arising from the adjoint crystal B ad, ℓ of level ℓ.