Dual graded graphs for Kac-Moody algebras

TL;DR

This paper introduces dual graded graphs associated with Kac-Moody algebras, extending combinatorial structures like insertion algorithms and analyzing their subposets, inspired by affine Schubert calculus.

Contribution

It constructs dual graded graphs for any Kac-Moody algebra, generalizing strong and weak orders, and develops folded insertion methods including shifted insertion as a special case.

Findings

Constructed dual graded graphs for arbitrary Kac-Moody algebras.

Defined folded insertion algorithms generalizing classical insertions.

Analyzed subgraphs as distributive posets, connecting to combinatorial representation theory.

Abstract

Motivated by affine Schubert calculus, we construct a family of dual graded graphs $(\Gamma_s,\Gamma_w)$ for an arbitrary Kac-Moody algebra $\g(A)$. The graded graphs have the Weyl group $W$ of $\g(A)$ 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 Stembridge, we analyze the induced subgraphs of $(\Gamma_s,\Gamma_w)$ which are distributive posets.