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Kirillov–Reshetikhin Crystals B1, s for \(\widehat {\mathfrak {s}\mathfrak {l}}_{n}\) Using Nakajima Monomials

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Abstract

We give a realization of the Kirillov–Reshetikhin crystal B1, s using Nakajima monomials for \(\widehat {\mathfrak {s}\mathfrak {l}}_{n}\) using the crystal structure given by Kashiwara. We describe the tensor product \(\bigotimes _{i=1}^{N} B^{1,s_{i}}\) in terms of a shift of indices, allowing us to recover the Kyoto path model. Additionally, we give a model for the KR crystals Br,1 using Nakajima monomials.

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Acknowledgements

The authors would like to thank Peter Tingley, Rinat Kedem, and Bolor Turmunkh for valuable discussions. The authors would like to thank Masato Okado, Ben Salisbury, and Anne Schilling for comments on earlier drafts of this paper. The authors thank the anonymous referee for many useful comments and improvements to this paper. This work benefited from computations using SageMath [10, 12].

The majority of this work was done while the authors were at the University of Minnesota.

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Emily Gunawan

  2. School of Mathematics and Physics, The University of Queensland, St. Lucia, QLD, 4072, Australia

    Travis Scrimshaw

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  1. Emily Gunawan
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  2. Travis Scrimshaw
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Correspondence to Emily Gunawan.

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Presented by: Peter Littelmann

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The authors were partially supported by the National Science Foundation RTG grant NSF/DMS-1148634.

Appendix: Examples with SageMath

Appendix: Examples with SageMath

We give some examples using SageMath [12] using the crystal of Nakajima monomials implemented by Ben Salisbury and Arthur Lubovsky.

We construct B1,2 for \(\widehat {\mathfrak {s}\mathfrak {l}}_{5}\) using Nakajima monomials and then compare with the tensor product with B1,1, verifying Theorem 4 in this case:

figure b

Next we construct B1,1 ⊗B1,2 for \(\widehat {\mathfrak {s}\mathfrak {l}}_{3}\):

figure c

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Gunawan, E., Scrimshaw, T. Kirillov–Reshetikhin Crystals B1, s for \(\widehat {\mathfrak {s}\mathfrak {l}}_{n}\) Using Nakajima Monomials. Algebr Represent Theor 23, 1–27 (2020). https://doi.org/10.1007/s10468-019-09904-5

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