Dual graded graphs for Kac–Moody algebras

Lam, Thomas; Shimozono, Mark

doi:10.2140/ant.2007.1.451

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Dual graded graphs for Kac–Moody algebras

Thomas F. Lam and Mark Shimozono

Vol. 1 (2007), No. 4, 451–488

DOI: 10.2140/ant.2007.1.451

Abstract

Motivated by affine Schubert calculus, we construct a family of dual graded graphs $(Γ_{s}, Γ_{w})$ for an arbitrary Kac–Moody algebra $g$ . The graded graphs have the Weyl group $W$ of $g e h$ 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 $(Γ_{s}, Γ_{w})$ which are distributive posets.

Keywords

dual graded graphs, Schensted insertion, affine insertion

Mathematical Subject Classification 2000

Primary: 05E10

Secondary: 57T15, 17B67

Milestones

Received: 28 March 2007

Revised: 4 August 2007

Accepted: 1 September 2007

Published: 1 November 2007

Authors

Thomas F. Lam
Department of Mathematics Harvard University Cambridge MA 02138 USA
http://www.math.harvard.edu/~tfylam/
Mark Shimozono
Department of Mathematics Virginia Polytechnic Institute and State University Blacksburg, VA 24061-0123 USA

Vol. 1, No. 4, 2007