|Statement||by Seok-Jin Kang and Hyeonmi Lee.|
|Series||RIMS -- 1584|
|Contributions||Kyōto Daigaku. Sūri Kaiseki Kenkyūjo.|
|LC Classifications||MLCSJ 2007/00032 (Q)|
|The Physical Object|
|Pagination||24 p. :|
|Number of Pages||24|
|LC Control Number||2007558269|
In , Kang introduced an affine combinatorial object called the Young wall, and showed that the crystal bases for the irreducible highest weight modules of level 1 for quantum affine algebras. further to construct crystal graphs of higher level irreducible highest weight modules for classical quantum aﬃne algebras excluding the D(1) n type . Under those constructions, the crystal graph of alevel-l irreducible highest weight module is described as l layers of level-1 Young walls and this is done using only the generic whole blocks. The constructions for each type were made in an ad hoc . We provide a unified approach to the Young wall description of crystal graphs for arbitrary level irreducible highest weight representations over classical quantum affine algebras. We provide a unified approach to the Young wall description of crystal graphs for arbitrary level irreducible highest weight representations over classical quantum affine algebras. The crystal graph is realized as the affine crystal consisting of all reduced Young walls built on a ground-state wall.
In the article Jung et al. () , we developed the combinatorics of Young walls associated with higher level adjoint crystals for the quantum aff. We develop the combinatorics of Young walls associated with higher level adjoint crystals for the quantum affine algebra Uq(sl2̂). The irreducible highest weight crystal B(λ)of arbitrary level is realized as the affine crystal consisting of reduced Young walls on λ. We also give a Young wall realization of the crystal B(∞)for Uq−(sl2̂). Cluster algebras and quantum affine algebras Hernandez, David and Leclerc, Bernard, Duke Mathematical Journal, ; Extremal weight modules of quantum affine algebras Nakajima, Hiraku,, ; Noncommutative projective curves and quantum loop algebras Schiffmann, Olivier, Duke Mathematical Journal, ; Symmetric quiver Hecke algebras and R -matrices of quantum affine algebras. The crystal bases for basic representations (i.e., highest weight representation of level 1) for quantum affine algebras are realized as the sets of reduced proper Young walls.
In, Kang and Lee extended his theory to the alization of the crystal basis for the irreducible highest weight representations of higher level for antum affine algebras. Also, in, Kang, Lee and the authors used Young walls to realize the ystal bases of irreducible highest weight modules over classical quantum finite algebras. In this paper we give a realization of crystal bases for quantum affine algebras using some new combinatorial objects which we call the Young walls. The Young walls consist of colored blocks with various shapes that are built on a given ground-state wall and can be viewed as generalizations of Young diagrams. The rules for building Young walls and the action of Kashiwara operators are given explicitly in terms of combinatorics of Young walls. resentations of the quantum aﬃne algebra Uq(C (1) 2) using some new combinatorial objects which we call the Young walls. The Young walls consist of colored blocks that are built on the given ground-state and can be viewed as generalizations of ∗This research was supported by KOSEF Grant # L and the Young Scientist. The Young walls, introduced by Kang in, provide a combinatorial model for level 1 highest weight crystals over quantum aﬃne algebras of classical Mathematics Subject Classiﬁcation. 17B37, 17B67, 20G42, 81R