# A uniform model for Kirillov―Reshetikhin crystals

Abstract : We present a uniform construction of tensor products of one-column Kirillov–Reshetikhin (KR) crystals in all untwisted affine types, which uses a generalization of the Lakshmibai–Seshadri paths (in the theory of the Littelmann path model). This generalization is based on the graph on parabolic cosets of a Weyl group known as the parabolic quantum Bruhat graph. A related model is the so-called quantum alcove model. The proof is based on two lifts of the parabolic quantum Bruhat graph: to the Bruhat order on the affine Weyl group and to Littelmann's poset on level-zero weights. Our construction leads to a simple calculation of the energy function. It also implies the equality between a Macdonald polynomial specialized at $t=0$ and the graded character of a tensor product of KR modules.
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Conference papers

https://hal.inria.fr/hal-01229719
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Submitted on : Tuesday, November 17, 2015 - 10:20:32 AM
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• HAL Id : hal-01229719, version 1

### Citation

Cristian Lenart, Satoshi Naito, Daisuke Sagaki, Anne Schilling, Mark Shimozono. A uniform model for Kirillov―Reshetikhin crystals. 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), 2013, Paris, France. pp.25-36. ⟨hal-01229719⟩

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