The graph form of the Combinatorial Invariance Conjecture

Let $W$ be a Coxeter group and let $u,v\in W$ with $u\leq v$. Let $B([u,v])$ denote the directed Bruhat graph of the interval $[u,v]$, and let $\widetilde{R}_{u,v}(q)$ be the polynomial defined by

$$R_{u,v}(q)=q^{\frac{\ell(v)-\ell(u)}{2}}\widetilde{R}_{u,v}(q^{\frac{1}{2}}-q^{-\frac{1}{2}}).$$

Graph Combinatorial Invariance conjecture. The $\widetilde{R}$-polynomial $\widetilde{R}_{u,v}(q)$ depends only on the isomorphism type of the graph $B([u,v])$.

Dyer’s result identifies the relevant directed Bruhat graph from the interval poset, making this conjecture equivalent to the Combinatorial Invariance Conjecture. The paper establishes corresponding congruence results and finite-length cases, but the general assertion remains open.