Dual Filtered Graphs for Kac-Moody algebras

We construct a strong filtered graph $\Gamma s(\Lambda )$ dependent on the dominant weight $\Lambda$, and a weak filtered graph $\Gamma w(\backslash Kcen)$ dependent on the canonical central element $\backslash Kcen$ for an arbitrary Kac-Moody algebra $g$. In our construction, both graphs $(\Gamma s(\Lambda ),\Gamma w(\backslash Kcen))$ have the vertex set as the Weyl group of $g$, with the grading given by the length function. The edges of the graph $\Gamma s(\backslash La)$ are labeled versions of the $\lambda$-chain model of K-Chevalley rules for Kac-Moody flag manifolds as developed by Lenart and Shimozono, originally defined by Lenart and Postnikov. Meanwhile, the labels on $\Gamma w(\backslash Kcen)$ come from the dual multiplication map of K-cohomology of affine Grassmannian $GrG$. We conjecture that the strong filtered graph and weak filtered graph are dual, which means we get an identity when we apply the up and down operators on the vertices. We proved this identity except one case that where we call the chain is $j$-present. Our identity is similar to the Möbius construction of the dual filtered graph, as previously studied by Patrias and Pylyavskyy, and in fact, in the limit $n\to \infty$ of the $An-1(1)$, our construction recovers their identity. We also expect to recover their combinatorics of Möbius deformation of the shifted Young's lattice in type $Cn(1)$ as $n$ approaches infinity.