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Dual graded graphs for Kac-Moody algebras. (English) Zbl 1200.05249
Summary: Motivated by affine Schubert calculus, we construct a family of dual graded graphs $$(\Gamma_s,\Gamma_w)$$for an arbitrary Kac-Moody algebra $$g$$. The graded graphs have the Weyl group $$W$$ of $$geh$$ 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 $$(\Gamma_s,\Gamma_w)$$ which are distributive posets.

##### MSC:
 05E10 Combinatorial aspects of representation theory 57T15 Homology and cohomology of homogeneous spaces of Lie groups 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 57M15 Relations of low-dimensional topology with graph theory
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