Perfect crystals of quantum affine Lie algebras

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

doi:10.1215/s0012-7094-92-06821-9

Cited by 194 publications

(317 citation statements)

References 11 publications

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“…N following [Kac], and crystals of U q (g) (or U ′ q (g)) 3 following [KMN1,KMN2]. We consider two categories of representations and crystals; one for the integrable highest weight modules and the other for finite-dimensional modules.…”

Section: Contents Of the Papermentioning

confidence: 99%

Paths, Crystals and Fermionic Formulae

Hatayama

Kuniba

Okado

et al. 2002

MathPhys Odyssey 2001

126

286

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We introduce a fermionic formula associated with any quantum affine algebra Uq(X (r) N ). Guided by the interplay between corner transfer matrix and Bethe ansatz in solvable lattice models, we study several aspects related to representation theory, most crucially, the crystal basis theory. They include one dimensional sums over both finite and semi-infinite paths, spinon character formulae, Lepowski-Primc type conjectural formula for vacuum string functions, dilogarithm identities, Q-systems and their solution by characters of various classical subalgebras and so forth. The results expand [HKOTY1] including the twisted cases and more details on inhomogeneous paths consisting of non-perfect crystals. As a most intriguing example, certain inhomogeneous one dimensional sums conjecturally give rise to branching functions of an integrable G

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Section: Contents Of the Papermentioning

confidence: 99%

Paths, Crystals and Fermionic Formulae

Hatayama

Kuniba

Okado

et al. 2002

MathPhys Odyssey 2001

126

286

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“…Originally the KKR bijection is defined in a purely combinatorial way, and it has no representation theoretic interpretation for a long time. Therefore it is expected that our algebraic reformulation will give some new insights into the theory of crystals for finite-dimensional representations of quantum affine Lie algebras [23][24][25].…”

Section: Introductionmentioning

confidence: 99%

Crystal interpretation of Kerov–Kirillov–Reshetikhin bijection II. Proof for $\mathfrak{sl}_{n}$ case

Sakamoto

2007

J Algebr Comb

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In proving the Fermionic formulae, a combinatorial bijection called the Kerov-Kirillov-Reshetikhin (KKR) bijection plays the central role. It is a bijection between the set of highest paths and the set of rigged configurations. In this paper, we give a proof of crystal theoretic reformulation of the KKR bijection. It is the main claim of Part I written by A. Kuniba, M. Okado, T. Takagi, Y. Yamada, and the author. The proof is given by introducing a structure of affine combinatorial R matrices on rigged configurations.Keywords Fermionic formulae · Kerov-Kirillov-Reshetikhin bijection · Rigged configuration · Crystal bases of quantum affine Lie algebras · Box-ball systems · Ultradiscrete soliton systems

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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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Perfect crystals of quantum affine Lie algebras

Cited by 194 publications

References 11 publications

Paths, Crystals and Fermionic Formulae

Paths, Crystals and Fermionic Formulae

Crystal interpretation of Kerov–Kirillov–Reshetikhin bijection II. Proof for $\mathfrak{sl}_{n}$ case

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

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