Abstract
There is a remarkable formula for the principal specialization of a type A Schubert polynomial as a weighted sum over reduced words. Taking appropriate limits transforms this to an identity for the backstable Schubert polynomials recently introduced by Lam, Lee, and Shimozono. This note identifies some analogues of the latter formula for principal specializations of Schubert polynomials in classical types B, C, and D. We also describe some more general identities for Grothendieck polynomials. As a related application, we derive a simple proof of a pipe dream formula for involution Grothendieck polynomials.
Highlights
There is a remarkable formula for the principal specialization Sw(1, q, q2, . . . , qn−1) of a Schubert polynomial as a weighted sum over reduced words
In this note we identify some apparently new analogues of Macdonald’s identity for the principal specializations of Schubert polynomials in other classical types
If we substitute xi → zi for i > 0 and xi → x1−i for i 0, SBw and SCw specialize to the Schubert polynomials of types B and C defined by Billey and Haiman in [1]; compare our definition with [1, Thm. 3]
Summary
There is a remarkable formula for the principal specialization Sw(1, q, q2, . . . , qn−1) of a (type A) Schubert polynomial as a weighted sum over reduced words. Let Reduced±B(w) and Reduced±C (w) denote the respective sets of signed reduced words for w. If we substitute xi → zi for i > 0 and xi → x1−i for i 0, SBw and SCw specialize to the Schubert polynomials of types B and C defined by Billey and Haiman in [1]; compare our definition with [1, Thm. 3]. If we again substitute xi → zi for i > 0 and xi → x1−i for i 0, our definition of the power series SDw specializes to Billey and Haiman’s formula for the Schubert polynomial of type D given in [1, Thm. 4]. Setting q = 1 in Theorem 1.5 leads to surprising enumerative formulas involving reduced words, compatible sequences, and plane partitions [5]. We are able to prove a more general K-theoretic formula, partially resolving an open question from [8, § 6]
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