Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.10276 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911671087267840 |
|---|---|
| author | Bai, Haojun Gu, Feng Guo, Peter L. Liu, Jiaji |
| author_facet | Bai, Haojun Gu, Feng Guo, Peter L. Liu, Jiaji |
| contents | Motivated by Stanley's ``Schubert shenanigans'' paper, commendable attempts have been made to understand the principal specializations of Schubert or Grothendieck polynomials. In this paper, we prove that when a permutation $w$ does not contain the $1423$ pattern, the principal specialization of the corresponding $β$-Grothendieck polynomial can be expressed nonnegatively in terms of the occurrences of patterns in $w$. Using an inverse conservation principle, we further obtain the nonnegativity expansion for permutations avoiding the $1342$ pattern. Our results partially resolve conjectures raised respectively by Gao (independently observed by Gaetz), Meśzáros--Tanjaya, and Dennin. The proofs are achieved based upon a reduction algorithm performing on the classic pipe dream model of $β$-Grothendieck polynomials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_10276 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Principal specializations of Grothendieck polynomials Bai, Haojun Gu, Feng Guo, Peter L. Liu, Jiaji Combinatorics Motivated by Stanley's ``Schubert shenanigans'' paper, commendable attempts have been made to understand the principal specializations of Schubert or Grothendieck polynomials. In this paper, we prove that when a permutation $w$ does not contain the $1423$ pattern, the principal specialization of the corresponding $β$-Grothendieck polynomial can be expressed nonnegatively in terms of the occurrences of patterns in $w$. Using an inverse conservation principle, we further obtain the nonnegativity expansion for permutations avoiding the $1342$ pattern. Our results partially resolve conjectures raised respectively by Gao (independently observed by Gaetz), Meśzáros--Tanjaya, and Dennin. The proofs are achieved based upon a reduction algorithm performing on the classic pipe dream model of $β$-Grothendieck polynomials. |
| title | Principal specializations of Grothendieck polynomials |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.10276 |