Consider the root system $R$ for a Coxeter system $(W,S)$ of type $A\_n$ with a choice of simple roots. Denote by $I(w)$ for $w\in W$ the set of positive roots $\beta\in R^+$ such that $w(\beta)$ is a negative root.


A subset $A\subseteq I(w)$ is said to be closed if for each $\beta,\gamma\in A$, if $\beta+\gamma\in I(w)$, then $\beta+\gamma\in A$.


We consider the roots as linear polynomials generating a polynomial ring. Consider the set $P\_w$ of rational functions (which are in fact reciprocals of polynomials) of the form
$$\prod\_{\beta\in A}\frac 1{\beta}$$
where $A\subseteq I(w)$ ranges over all closed subsets of $I(w)$. Is $P\_w$ linearly independent over $\mathbb{Z}$?


This arose because I want to determine linear independence of certain elements of the nil-Hecke ring.



Linear independence of reciprocals of products of closed sets of roots in type $A$ inversion sets
I found a counterexample. Every subset of $I(s\_2s\_3s\_1s\_2)$ is closed, but there is a dependence relation for subsets of size $3$, where if we clear denominators we get
$$(\alpha\_1+\alpha\_2)-\alpha\_2+(\alpha\_2+\alpha\_3)-(\alpha\_1+\alpha\_2+\alpha\_3)=0$$


