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The genomic Schur function is fundamental-positive. (English) Zbl 1435.05230

Summary: O. Pechenik and A. Yong [Forum Math. Pi 5, Article ID e3, 128 p. (2017; Zbl 1369.14060)] 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 [“A note on passing from a quasi-symmetric function expansion to a Schur function expansion of a symmetric function”, Preprint, arXiv:1802.09686], this yields a compact combinatorial (but necessarily nonpositive) formula for the Schur expansion of \(U_\lambda\).

MSC:

05E10 Combinatorial aspects of representation theory
05E14 Combinatorial aspects of algebraic geometry
05E05 Symmetric functions and generalizations
14M15 Grassmannians, Schubert varieties, flag manifolds

Citations:

Zbl 1369.14060
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References:

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