Motivated by affine Schubert calculus, we construct a family of dual graded graphs for an arbitrary Kac–Moody algebra . The graded graphs have the Weyl group of as vertex set and are labeled versions of the strong and weak orders of 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 Proctor and Stembridge, we analyze the induced subgraphs of which are distributive posets.
"Dual graded graphs for Kac–Moody algebras." Algebra Number Theory 1 (4) 451 - 488, 2007. https://doi.org/10.2140/ant.2007.1.451