#### Vol. 1, No. 4, 2007

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Dual graded graphs for Kac–Moody algebras

### Thomas F. Lam and Mark Shimozono

Vol. 1 (2007), No. 4, 451–488
##### 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.

##### Keywords
dual graded graphs, Schensted insertion, affine insertion
##### Mathematical Subject Classification 2000
Primary: 05E10
Secondary: 57T15, 17B67