Schubert polynomials, slide polynomials, Stanley symmetric functions and quasi-Yamanouchi pipe dreams

Abstract

We introduce two new bases for polynomials that lift monomial and fundamental quasisymmetric functions to the full polynomial ring. By defining a new condition on pipe dreams, called quasi-Yamanouchi, we give a positive combinatorial rule for expanding Schubert polynomials into these new bases that parallels the expansion of Schur functions into fundamental quasisymmetric functions. As a result, we obtain a refinement of the stable limits of Schubert polynomials to Stanley symmetric functions. We also give combinatorial rules for the positive structure constants of these bases that generalize the quasi-shuffle product and shuffle product, respectively. We use this to give a Littlewood--Richardson rule for expanding a product of Schubert polynomials into fundamental slide polynomials and to give formulas for products of Stanley symmetric functions in terms of Schubert structure constants.

28 pages

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• Flagged $(\mathcal{P},\rho )$-partitions
• Asymmetric function theory
• The genomic Schur function is fundamental-positive
• On the Schur expansion of Jack polynomials
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• Enumerating quasi-Yamanouchi tableaux of Durfee size two
• On formulas and some combinatorial properties of Schubert Polynomials
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• Proof of a $K$-theoretic polynomial conjecture of Monical, Pechenik, and Searles