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Discrete Mathematics & Theoretical Computer Science |
Cristian Lenart ; Satoshi Naito ; Daisuke Sagaki ; Anne Schilling ; Mark Shimozono - A uniform model for Kirillov―Reshetikhin crystals
dmtcs:12790 - Discrete Mathematics & Theoretical Computer Science, January 1, 2013, DMTCS Proceedings vol. AS, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013) - https://doi.org/10.46298/dmtcs.12790- 1 Department of Mathematics and Statistics [Albany-USA]
- 2 Department of Mathematics [Tokyo]
- 3 Institute of Mathematics, University of Tsukuba
- 4 Department of Mathematics [Univ California Davis]
- 5 Department of Mathematics [Blacksburg]
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.