Affine Crystals and Vertex Models

Kang, Seok-Jίn; Kashiwara, Masaki; Misra, Kailash C.; Miwa, Tetsuji; Nakashima, Torahiko; Nakayashiki, Atsushi

doi:10.1142/s0217751x92003896

Cited by 248 publications

(450 citation statements)

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“…In order to understand the representations of g 0 when g 0 is of affine type, Kang adopted the path realization of crystal basis ( [KMN1,KMN2]) for affine KacMoody algebras in [Ka3,Ka4,KM1]. This idea was followed by Klima and Misra [KMi] for the indefinite Kac-Moody algebras of symplectic type.…”

Section: Kang's Generalizationsmentioning

confidence: 99%

Dimensions of imaginary root spaces of hyperbolic Kac–Moody algebras

Carbone¹,

Freyn²,

Lee³

2014

Contemporary Mathematics

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Section: Kang's Generalizationsmentioning

confidence: 99%

Dimensions of imaginary root spaces of hyperbolic Kac–Moody algebras

Carbone¹,

Freyn²,

Lee³

2014

Contemporary Mathematics

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“…al. in [10] and [11] (see [6] for a more recent explanation). We do this by exhibiting an explicit crystal isomorphism between the highest irreducible component of the abacus model and the Kyoto path model for a particular perfect crystal and ground state path.…”

Section: Introductionmentioning

confidence: 98%

“…Section 4.2 is independent of the rest of the paper, so the rest of our results remain true as originally stated. The bead string corresponding to the partition (12,11,10,9,7,5,3,3,3,1). We fix the origin to be the vertex of the partition, so the beads are in positions .…”

Section: Introductionmentioning

confidence: 99%

Three Combinatorial Models for $$ _ $$ Crystals, with Applications to Cylindric Plane Partitions

Tingley¹

2010

International Mathematics Research Notices

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Abstract. We define three combinatorial models for b sln crystals, parametrized by partitions, configurations of beads on an "abacus", and cylindric plane partitions, respectively. These are reducible, but we can identify an irreducible subcrystal corresponding to any dominant integral highest weight Λ. Cylindric plane partitions actually parametrize a basis for V Λ ⊗ F , where F is the space spanned by partitions. We use this to calculate the partition function for a system of random cylindric plane partitions. We also observe a form of rank level duality. Finally, we use an explicit bijection to relate our work to the Kyoto path model.

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“…n , in [15], Kim gave an explicit description and she gave crystal isomorphisms from monomial realization to path realization and Young wall realization given in [3,5,6,7].…”

Section: Introductionmentioning

confidence: 99%