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Genomic tableaux

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Abstract

We explain how genomic tableaux [Pechenik–Yong ’15] are a semistandard complement to increasing tableaux [Thomas–Yong ’09]. From this perspective, one inherits genomic versions of jeu de taquin, Knuth equivalence, infusion and Bender–Knuth involutions, as well as Schur functions from (shifted) semistandard Young tableaux theory. These are applied to obtain new Littlewood–Richardson rules for K-theory Schubert calculus of Grassmannians (after [Buch ’02]) and maximal orthogonal Grassmannians (after [Clifford–Thomas–Yong ’14], [Buch–Ravikumar ’12]). For the unsolved case of Lagrangian Grassmannians, sharp upper and lower bounds using genomic tableaux are conjectured.

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Notes

  1. The genomic analogy is that boxes of a gene are alleles, and the other genes of the same family are paralogs.

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Acknowledgments

We thank Hugh Thomas for many conversations which helped to make this project possible. We thank the referees for helpful comments that improved the exposition of this paper. OP was supported by an NSF Graduate Research Fellowship, and Illinois Distinguished Fellowship from the University of Illinois, and NSF MCTP grant DMS 0838434. AY was supported by NSF grants and a Helen Corley Petit fellowship at UIUC.

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Authors and Affiliations

  1. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854, USA

    Oliver Pechenik

  2. Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, IL, 61801, USA

    Alexander Yong

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  1. Oliver Pechenik
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  2. Alexander Yong
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Correspondence to Oliver Pechenik.

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Pechenik, O., Yong, A. Genomic tableaux. J Algebr Comb 45, 649–685 (2017). https://doi.org/10.1007/s10801-016-0720-8

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