Rigged configurations for all symmetrizable types

TL;DR

This paper extends the rigged configuration model to all symmetrizable types, providing a universal combinatorial tool for crystal bases and Littlewood-Richardson rules across Kac-Moody types.

Contribution

The authors generalize the rigged configuration model to all symmetrizable types, broadening its applicability beyond finite and affine types.

Findings

Rigged configuration model valid for all symmetrizable types

Provides a universal Littlewood-Richardson rule

Simplifies combinatorial understanding of crystals

Abstract

In an earlier work, the authors developed a rigged configuration model for the crystal $B(\infty)$ (which also descends to a model for irreducible highest weight crystals via a cutting procedure). However, the result obtained was only valid in finite types, affine types, and simply-laced indefinite types. In this paper, we show that the rigged configuration model proposed does indeed hold for all symmetrizable types. As an application, we give an easy combinatorial condition that gives a Littlewood-Richardson rule using rigged configurations which is valid in all symmetrizable Kac-Moody types.