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Algorithm classifying roots of star-shaped Kac-Moody root systems

Toshio Oshima

TL;DR

The paper addresses classifying Weyl-group orbit representatives of roots with prescribed norm in star-shaped Kac-Moody root systems and provides an effective computer algorithm to produce them. It maps roots to spectral types of Fuchsian differential equations via tuples of partitions and the index of rigidity idx, and implements the core routine spbasic in Risa/Asir to enumerate fundamental tuples with a given idx. It also links these classifications to middle convolution and addition operations that realize Weyl-group actions, and to the theory of rigid and nonrigid Fuchsian systems and higher-dimensional Painlevé-type equations. The work thus bridges Lie-theoretic root data with spectral problems in differential equations, offering concrete computational tools for studying spectral types, rigidity, and isomonodromic deformations.

Abstract

For a star-shaped Kac-Moody root system, we provide an effective algorithm to obtain representatives of the Weyl group orbits of roots with a given norm and implement it as a computer program. We also explain the relationship between these orbits and Fuchsian differential equations on the Riemann sphere, as well as higher-dimensional Painleve-type equations.

Algorithm classifying roots of star-shaped Kac-Moody root systems

TL;DR

The paper addresses classifying Weyl-group orbit representatives of roots with prescribed norm in star-shaped Kac-Moody root systems and provides an effective computer algorithm to produce them. It maps roots to spectral types of Fuchsian differential equations via tuples of partitions and the index of rigidity idx, and implements the core routine spbasic in Risa/Asir to enumerate fundamental tuples with a given idx. It also links these classifications to middle convolution and addition operations that realize Weyl-group actions, and to the theory of rigid and nonrigid Fuchsian systems and higher-dimensional Painlevé-type equations. The work thus bridges Lie-theoretic root data with spectral problems in differential equations, offering concrete computational tools for studying spectral types, rigidity, and isomonodromic deformations.

Abstract

For a star-shaped Kac-Moody root system, we provide an effective algorithm to obtain representatives of the Weyl group orbits of roots with a given norm and implement it as a computer program. We also explain the relationship between these orbits and Fuchsian differential equations on the Riemann sphere, as well as higher-dimensional Painleve-type equations.

Paper Structure

This paper contains 4 sections, 3 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Fundamental tuples of partitions
  3. Algorithm obtaining fundamental tuples
  4. Related results

Key Result

Lemma 1.3

For the expression eq:root of $\alpha\in\Delta_+$ with $\mathop{\mathrm{supp}}\nolimits\alpha\varsupsetneqq\{\alpha_0\}$, we have Moreover $\alpha\in Q_+\setminus\{0\}$ is an element of $B$ if and only if Note that these two conditions correspond to $(\alpha|\alpha_{j,\nu})\le0$ and $(\alpha|\alpha_0)\le0$.

Theorems & Definitions (12)

  • Remark 1.1: Kc
  • Remark 1.2
  • Lemma 1.3: Orims
  • Remark 1.4
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Remark 2.5
  • Example 2.6
  • ...and 2 more