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arXiv · 2610.32832

Kazhdan--Lusztig Polynomials for permutations with $P_{\mathrm{id},w}(1)=3$ and Irreducible Singular Locus

Gaston Burrull·Zhitong Alex Li·Shixuan Zeng
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Abstract

We determine the Kazhdan--Lusztig polynomial $P_{\mathrm{id},w}(q)$ for permutations $w$ such that $P_{\mathrm{id},w}(1)=3$ 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 $a 1$. This proves the irreducible singular locus case of a 2009 conjecture of Billey and Weed.

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Gaston Burrull, Zhitong Alex Li, Shixuan Zeng. 2026-10-23. Kazhdan--Lusztig Polynomials for permutations with $P_{\mathrm{id},w}(1)=3$ and Irreducible Singular Locus. https://arxiv.org/abs/2610.32832

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