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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.

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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