2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/138974Motivated by affine Schubert calculus, we construct a family of dual graded graphs $(Γ_s,Γ_w)$ for an arbitrary Kac-Moody algebra $\g(A)$. The graded graphs have the Weyl group $W$ of $\g(A)$ as vertex set and are labeled versions of the strong and weak orders of $W$ respectively. Using a construction of Lusztig for quivers with an admissible automorphism, we define folded insertion for a Kac-Moody algebra and obtain Sagan-Worley shifted insertion from Robinson-Schensted insertion as a special case. Drawing on work of Stembridge, we analyze the induced subgraphs of $(Γ_s,Γ_w)$ which are distributive posets.36 pagesCombinatoricsAlgebraic GeometryRepresentation Theory05E10; 57T15; 17B67text