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  urlcolor=MidnightBlue,pdftitle={Combinatorial invariance of Kazhdan--Lusztig polynomials},pdfauthor={OpenAI}}
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\title{Combinatorial invariance of\\Kazhdan--Lusztig polynomials}
\author{OpenAI}
\date{September 24, 2026}
\begin{document}
\maketitle
\begin{abstract}
Fix a Coxeter system $(W,S)$. For $u\le b$ in Bruhat order, the
Kazhdan--Lusztig polynomial $P_{u,b}(q)$ is defined by a recursion that
uses the simple generators of $W$; it also computes local intersection
cohomology of Schubert varieties. This paper shows that the polynomial
depends only on the abstract poset $[u,b]$. If $[u,b]$ and $[u',b']$
are isomorphic as abstract graded posets, then $P_{u,b}=P_{u',b'}$, even
when the ambient Coxeter systems are infinite or noncrystallographic.
The same conclusion holds for every subinterval, and reciprocity then
recovers all $R$-polynomials on the interval. The proof uses
Braden--MacPherson sheaves on the Bruhat moment graph and compares two
realizations through a reflection order transported by the poset
isomorphism.
\end{abstract}
\tableofcontents
\section{Introduction}\label{sec:introduction}
A \emph{Coxeter system} $(W,S)$ is a group generated by a set $S$ of
involutions subject to relations that dictate how the generators
multiply. The two most familiar examples are the symmetric groups $S_n$
(whose generators are the adjacent transpositions) and the dihedral
groups (symmetry groups of a regular polygon, generated by two
reflections). Coxeter groups occur throughout mathematics: they
classify finite reflection groups, they describe Schubert cells in flag
varieties, and they connect to Hecke algebras and quantum groups.
For a Coxeter system $(W,S)$, write $\ell$ for its length function and
$\le$ for Bruhat order. For $u\le b$, the interval
\[
 [u,b]=\{x\in W:u\le x\le b\}
\]
is a finite graded poset of rank $\ell(b)-\ell(u)$, even when $W$ is
infinite. Let $P^W_{u,b}(q)$ denote the usual equal-parameter
Kazhdan--Lusztig polynomial, normalized by $P^W_{u,u}=1$ and
$\deg P^W_{u,b}< (\ell(b)-\ell(u))/2$ for $u<b$.
The defining recursion for these polynomials uses the simple generators
of a Coxeter system, while geometric and categorical constructions use
root labels. An abstract Bruhat interval poset retains neither kind of
data. The \emph{combinatorial invariance conjecture} asks whether
isomorphic interval posets have the same polynomial. An isomorphism here
is only a bijection preserving and reflecting order; it carries no
labels or root data. The following theorem gives a positive answer in
full equal-parameter generality.
\begin{theorem}[Combinatorial invariance]\label{thm:main}
Let $(W,S)$ and $(W',S')$ be arbitrary Coxeter systems, and let
$u\le b$ in $W$ and $u'\le b'$ in $W'$.
If
\[
 \iota:[u,b]\longrightarrow[u',b']
\]
is an isomorphism of abstract Bruhat interval posets, then their
equal-parameter Kazhdan--Lusztig polynomials satisfy
\[
 P^W_{u,b}(q)=P^{W'}_{u',b'}(q).
\]
\end{theorem}
The statement includes infinite and noncrystallographic groups.
Restricting $\iota$ to subintervals also determines every
polynomial $P_{x,y}$ on the interval; reciprocity then recovers all its
$R$-polynomials (Corollary~\ref{cor:all-subintervals}).