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arXiv · 1810.04727

The genomic Schur function is fundamental-positive

Oliver Pechenik
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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 non-trivially 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 non-positive) formula for the Schur expansion of $U_\lambda$.

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Oliver Pechenik. 2018-10-10. The genomic Schur function is fundamental-positive. https://doi.org/10.1007/s00026-019-00483-2

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