Dual graded graphs for Kac–Moody algebras

Abstract

Motivated by affine Schubert calculus, we construct a family of dual graded graphs $\left({\Gamma }_s,{\Gamma }_w\right)$ for an arbitrary Kac–Moody algebra $\mathfrak{g}$. The graded graphs have the Weyl group $W$ of $\mathfrak{g}\mathfrak{e}\mathfrak{h}$ 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 Proctor and Stembridge, we analyze the induced subgraphs of $\left({\Gamma }_s,{\Gamma }_w\right)$ which are distributive posets.