Casian–Kodama conjecture on Toda blow-up graphs and flag-manifold cohomology

Let ${\mathcal G}$ be the graph associated with the blow-up structure of the Toda lattice, and let ${\mathcal B}$ (or $\hat{\mathcal B}$ in the affine case) be the corresponding real flag manifold, decomposed into Bruhat cells. For each sign choice $\epsilon$, let ${\mathcal G}_\epsilon$ be the corresponding graph and let ${\mathcal L}$ be the associated local system on the real flag manifold.

Casian–Kodama conjecture. The graph ${\mathcal G}$ is the graph of incidence numbers for the integral cohomology of the real flag manifold ${\mathcal B}$ ($\hat{\mathcal B}$) in terms of the Bruhat cells. In general, each $\epsilon$ corresponds to a local system ${\mathcal L}$ in the real flag manifold and the graph ${\mathcal G}_\epsilon$ is the graph of incidence numbers in the computation of integral cohomology with coefficients in ${\mathcal L}$.

The conjecture identifies the combinatorial graph determined by Toda blow-ups with the cellular incidence data computing integral cohomology, including cohomology with local coefficients.