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Crystal bases for quantum affine algebras and combinatorics of Young walls

Seok-Jin Kang
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Abstract

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 the 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. The crystal graphs for basic representations are characterized as the set of all reduced proper Young walls. The characters of basic representations can be computed easily by counting the number of colored blocks that have been added to the ground-state wall.

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Seok-Jin Kang. 2002-07-30. Crystal bases for quantum affine algebras and combinatorics of Young walls. https://arxiv.org/abs/math/0207283

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