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Kazhdan--Lusztig Polynomials for permutations with Pid,w(1)=3 and Irreducible Singular Locus

  • Gaston Burrull
  • Zhitong Alex Li
  • Shixuan Zeng
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Abstract

We determine the Kazhdan--Lusztig polynomial Pid,w(q) for permutations w such that Pid,w(1)=3 and the Schubert variety Xw has irreducible singular locus. Let h=h(w) be the minimum height $w(a)-w(d)$ of a $3412$-occurrence in w at positions a<b<c<d, with h=1 when no such occurrence exists. We show that Pid,w(q)=1+qh+qh+1 if exactly one of the patterns $3412$ and $4231$ occurs, and Pid,w(q)=1+q+qh if both patterns occur. In the latter case, h>1. This proves the irreducible singular locus case of a 2009 conjecture of Billey and Weed.

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Cite

BibTeXRIS
BibTeX
@misc{burrull2026kazhdanlusztig,
  title = {{Kazhdan--Lusztig Polynomials for permutations with $P_\mathrmid,w(1)=3$ and Irreducible Singular Locus}},
  author = {Gaston Burrull and Zhitong Alex Li and Shixuan Zeng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.32832},
  eprint = {2610.32832},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.32832},
}
RIS
TY  - UNPB
TI  - Kazhdan--Lusztig Polynomials for permutations with P_id,w(1)=3 and Irreducible Singular Locus
AU  - Gaston Burrull
AU  - Zhitong Alex Li
AU  - Shixuan Zeng
PY  - 2026
DA  - 2026/10/23
PB  - arXiv
DO  - 10.48550/arxiv.2610.32832
UR  - https://arxiv.org/abs/2610.32832
AB  - We determine the Kazhdan--Lusztig polynomial P_id,w(q) for permutations w such that P_id,w(1)=3 and the Schubert variety X_w has irreducible singular locus. Let h=h(w) be the minimum height $w(a)-w(d)$ of a $3412$-occurrence in w at positions a<b<c<d, with h=1 when no such occurrence exists. We show that P_id,w(q)=1+q^h+q^h+1 if exactly one of the patterns $3412$ and $4231$ occurs, and P_id,w(q)=1+q+q^h if both patterns occur. In the latter case, h>1. This proves the irreducible singular locus case of a 2009 conjecture of Billey and Weed.
ER  -