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Pieri and Littlewood–Richardson rules for two rows and cluster algebra structure

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Abstract

We study an algebra encoding a twice-iterated Pieri rule for the representations of the general linear group and prove that it has the structure of a cluster algebra. We also show that its cluster variables invariant under a unipotent subgroup generate the highest weight vectors of irreducible representations occurring in the decomposition of the tensor product of two irreducible representations of the general linear group one of whom is labeled by a Young diagram with less than or equal to two rows.

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Acknowledgements

We thank Hyun Kyu Kim and Kyungyong Lee for helpful discussions. This work was supported by a Korea University New Faculty Start-up Grant.

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Authors and Affiliations

  1. Department of Mathematics, Korea University, Seoul, 02841, South Korea

    Sangjib Kim

  2. Department of Mathematics, University of Rochester, Rochester, NY, 14627, USA

    Semin Yoo

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  1. Sangjib Kim
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  2. Semin Yoo
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Correspondence to Sangjib Kim.

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Kim, S., Yoo, S. Pieri and Littlewood–Richardson rules for two rows and cluster algebra structure. J Algebr Comb 45, 887–909 (2017). https://doi.org/10.1007/s10801-016-0728-0

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  • DOI: https://doi.org/10.1007/s10801-016-0728-0

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