Geodesic
Schubert polynomial expansions revisited
Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e106

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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 $\sum Y_i\partial _i=\operatorname {id}$ 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. 
Nadeau, Philippe; Spink, Hunter; Tewari, Vasu. Schubert polynomial expansions revisited. Forum of Mathematics, Sigma, Tome 13 (2025) no. 1, p. e106. doi: 10.1017/fms.2025.10068
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