Dual graded graphs for Kac-Moody algebras. (English) Zbl 1200.05249

Lam, Thomas F.; Shimozono, Mark

Algebra Number Theory 1, No. 4, 451-488 (2007).

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

Keywords:

dual graded graphs; Robinson-Schensted insertion; Sagan-Worley insertion; affine insertion