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The Genomic Schur Function is Fundamental-Positive

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Abstract

In work with A. Yong, the author introduced genomic tableaux to prove the first positive combinatorial rule for the Littlewood–Richardson coefficients in torus-equivariant K-theory of Grassmannians. We then studied the genomic Schur function \(U_\lambda \), a generating function for such tableaux, showing that it is nontrivially a symmetric function, although generally not Schur-positive. Here, we show that \(U_\lambda \) is, however, positive in the basis of fundamental quasisymmetric functions. We give a positive combinatorial formula for this expansion in terms of gapless increasing tableaux; this is, moreover, the first finite expression for \(U_\lambda \). Combined with work of A. Garsia and J. Remmel, this yields a compact combinatorial (but necessarily nonpositive) formula for the Schur expansion of \(U_\lambda \).

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Acknowledgements

This paper was inspired by conversations with Bruce Westbury during the conference “SageDays@ICERM: Combinatorics and Representation Theory,” held July 2018 at the Institute for Computational and Experimental Research in Mathematics. Thanks to the organizers (Gabriel Feinberg, Darij Grinberg, Ben Salisbury, and Travis Scrimshaw) for creating such a productive environment. The author is also grateful for helpful conversations with Dominic Searles, Emily Sergel, and David Speyer. The author would also like to thank two anonymous referees for careful reading and many helpful comments.

The author was supported by a Mathematical Sciences Postdoctoral Research Fellowship (#1703696) from the National Science Foundation.

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  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    O. Pechenik

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Pechenik, O. The Genomic Schur Function is Fundamental-Positive. Ann. Comb. 24, 95–108 (2020). https://doi.org/10.1007/s00026-019-00483-2

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