Zigzag strip bundles and crystals

Jeong Ah Kim; Dong Uy Shin

doi:10.1016/j.jcta.2013.02.007

Zigzag strip bundles and crystals

Jeong Ah Kim, Dong Uy Shin

Department of Mathematics

Hanyang University

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

We introduce new combinatorial models, called zigzag strip bundles, over quantum affine algebras U_q(B_n⁽¹⁾), U_q(D_n⁽¹⁾) and U_q(D_n+1⁽²⁾), and show that the sets of all zigzag strip bundles for U_q(B_n⁽¹⁾), U_q(D_n⁽¹⁾) and U_q(D_n+1⁽²⁾) realize the crystal bases B( ∞ ) of U_q^-(B_n⁽¹⁾), U_q^-(D_n⁽¹⁾) and U_q^-(D_n+1⁽²⁾), respectively. Further, we discuss the connection between zigzag strip bundle realization, Nakajima monomial realization, and polyhedral realization of the crystal B( ∞ ).

Original language	English
Pages (from-to)	1087-1115
Number of pages	29
Journal	Journal of Combinatorial Theory. Series A
Volume	120
Issue number	5
DOIs	https://doi.org/10.1016/j.jcta.2013.02.007
State	Published - Jul 2013

Keywords

Crystals
Kashiwara embeddings
Nakajima monomials
Zigzag strip bundles

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10.1016/j.jcta.2013.02.007

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title = "Zigzag strip bundles and crystals",

abstract = "We introduce new combinatorial models, called zigzag strip bundles, over quantum affine algebras Uq(Bn(1)), Uq(Dn(1)) and Uq(Dn+1(2)), and show that the sets of all zigzag strip bundles for Uq(Bn(1)), Uq(Dn(1)) and Uq(Dn+1(2)) realize the crystal bases B( ∞ ) of Uq-(Bn(1)), Uq-(Dn(1)) and Uq-(Dn+1(2)), respectively. Further, we discuss the connection between zigzag strip bundle realization, Nakajima monomial realization, and polyhedral realization of the crystal B( ∞ ).",

keywords = "Crystals, Kashiwara embeddings, Nakajima monomials, Zigzag strip bundles",

author = "Kim, {Jeong Ah} and Shin, {Dong Uy}",

year = "2013",

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T1 - Zigzag strip bundles and crystals

AU - Kim, Jeong Ah

AU - Shin, Dong Uy

PY - 2013/7

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N2 - We introduce new combinatorial models, called zigzag strip bundles, over quantum affine algebras Uq(Bn(1)), Uq(Dn(1)) and Uq(Dn+1(2)), and show that the sets of all zigzag strip bundles for Uq(Bn(1)), Uq(Dn(1)) and Uq(Dn+1(2)) realize the crystal bases B( ∞ ) of Uq-(Bn(1)), Uq-(Dn(1)) and Uq-(Dn+1(2)), respectively. Further, we discuss the connection between zigzag strip bundle realization, Nakajima monomial realization, and polyhedral realization of the crystal B( ∞ ).

AB - We introduce new combinatorial models, called zigzag strip bundles, over quantum affine algebras Uq(Bn(1)), Uq(Dn(1)) and Uq(Dn+1(2)), and show that the sets of all zigzag strip bundles for Uq(Bn(1)), Uq(Dn(1)) and Uq(Dn+1(2)) realize the crystal bases B( ∞ ) of Uq-(Bn(1)), Uq-(Dn(1)) and Uq-(Dn+1(2)), respectively. Further, we discuss the connection between zigzag strip bundle realization, Nakajima monomial realization, and polyhedral realization of the crystal B( ∞ ).

KW - Crystals

KW - Kashiwara embeddings

KW - Nakajima monomials

KW - Zigzag strip bundles

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JO - Journal of Combinatorial Theory. Series A

JF - Journal of Combinatorial Theory. Series A

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Zigzag strip bundles and crystals

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