Abstract
In their work on the infinite flag variety, Lam, Lee, and Shimozono [30] introduced objects called bumpless pipe dreams and used them to give a formula for double Schubert polynomials. We extend this formula to the setting of K-theory, giving an expression for double Grothendieck polynomials as a sum over a larger class of bumpless pipe dreams. Our proof relies on techniques found in an unpublished manuscript of Lascoux [27]. Lascoux showed how to write double Grothendieck polynomials as a sum over alternating sign matrices. We explain how to view the Lam-Lee-Shimozono formula as a disguised special case of Lascoux's alternating sign matrix formula. Knutson, Miller, and Yong [21] gave a tableau formula for vexillary Grothendieck polynomials. We recover this formula by showing vexillary marked bumpless pipe dreams and flagged set-valued tableaux are in weight preserving bijection. Finally, we give a bijection between Hecke bumpless pipe dreams and decreasing tableaux. The restriction of this bijection to Edelman-Greene bumpless pipe dreams solves a problem of Lam, Lee, and Shimozono.
Original language | English (US) |
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Article number | 105470 |
Journal | Journal of Combinatorial Theory. Series A |
Volume | 182 |
DOIs | |
State | Published - Aug 2021 |
Externally published | Yes |
Bibliographical note
Publisher Copyright:© 2021 Elsevier Inc.
Keywords
- Alternating sign matrices
- Bumpless pipe dreams
- Grothendieck polynomials