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A combinatorial duality between the weak and strong Bruhat orders. (English) Zbl 1433.05319
Summary: C. Gaetz and Y. Gao [Proc. Am. Math. Soc. 148, No. 1, 1–7 (2020; Zbl 07144479)] used an order lowering operator $$\nabla$$, introduced by R. P. Stanley [Enumerative combinatorics. Vol. 1. 2nd ed. Cambridge: Cambridge University Press (2012; Zbl 1247.05003)], to prove the strong Sperner property for the weak Bruhat order on the symmetric group. Hamaker, Pechenik, Speyer, and Weigandt interpreted $$\nabla$$ as a differential operator on Schubert polynomials and used this to prove a new identity for Schubert polynomials and a determinant conjecture of Stanley. In this paper we study a raising operator $$\Delta$$ for the strong Bruhat order on the symmetric group, which is in many ways dual to $$\nabla$$. We prove a Schubert identity dual to that of Z. Hamaker et al. [“Derivatives of Schubert polynomials and proof of a determinant conjecture of Stanley”, Algebr. Comb. (to appear)] and derive formulas for counting weighted paths in the Hasse diagrams of the strong order which agree with path counting formulas for the weak order, providing a strong order analog of Macdonald’s reduced word identity. We also show that powers of $$\nabla$$ and $$\Delta$$ have the same Smith normal forms, which we describe explicitly, answering a question of Stanley.

##### MSC:
 05E10 Combinatorial aspects of representation theory 14N15 Classical problems, Schubert calculus
##### Keywords:
weak order; Bruhat order; Schubert polynomial; duality
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##### References:
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