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Schubert polynomial expansions revisited

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arxiv 2407.02375 v1 pith:LPFW46PR submitted 2024-07-02 math.CO

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keywords polynomialspolynomialslideexpansionsschubertdifferencedividedanalogue
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abstract

We give an elementary approach utilizing only the divided difference formalism for obtaining expansions of Schubert polynomials that are manifestly nonnegative, by studying solutions to the equation $\sum Y_i\partial_i=\mathrm{id}$ on polynomials with no constant term. This in particular recovers the pipe dream and slide polynomial expansions. We also show that slide polynomials satisfy an analogue of the divided difference formalisms for Schubert polynomials and forest polynomials, which gives a simple method for extracting the coefficients of slide polynomials in the slide polynomial decomposition of an arbitrary polynomial.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Cauchy identities for Grothendieck polynomials and a dual RSK correspondence through pipe dreams

    math.CO 2025-06 conditional novelty 7.0 of 10

    A new algorithm, pipe dream rectification, bijectively realizes the Cauchy identities for Grothendieck polynomials and leads to a pipe-dream version of dual RSK insertion.

  2. Equivariant quasisymmetry and noncrossing partitions

    math.CO 2025-04 conditional novelty 7.0 of 10

    A new equivariant version of quasisymmetric polynomials is introduced, giving double forest polynomials with noncrossing-partition evaluation formulas, Graham-positive structure constants, and a Bruhat-order character...

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