Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Classification of the root systems $R(m)$

Patrick Polo

TL;DR

This work completes the analysis of root subsystems $R(m)$ obtained from a reduced irreducible root system $R$ by selecting roots whose height is a multiple of $m$ (with Coxeter number $h$ and $2\le m<h$). By combining a height-partition approach with detailed case analysis across all root-system types, it identifies a Levi-type base $\,\Gamma(m)$ for $R(m)$ in almost all cases and isolates precise exceptional configurations where an extra root $\delta_{2m}$ appears and $d_m>1$. The results unify and extend prior work of Nadimpalli–Pattanayak–Prasad by giving explicit $\,\Gamma(m)$, the associated weight $\omega$ (often a minuscule fundamental weight), and the Weyl-dimension-derived constant $d_m$ for every type, including the exceptional types $E_6,E_7,E_8$ and the low-rank cases $F_4,G_2$. The findings sharpen the understanding of $R(m)$ in the context of torsion-element character theory and connect the subsystem structure to representations of the dual root system via $d_m$, with concrete tableaux and Dynkin-diagram constructions provided where needed. This classification aids precise computations of representation dimensions and clarifies when $R(m)$ behaves as a standard Levi-type subsystem versus when extra nodes alter the Levi structure.

Abstract

Let $R$ be a reduced irreducible root system, $h$ its Coxeter number and $m$ a positive integer smaller than $h$. Choose of base of $R$, whence a corresponding height function, and let $R(m)$ be the set of roots whose height is a multiple of $m$. In a recent paper, S. Nadimpalli, S. Pattanayak and D. Prasad studied, for the purposes of character theory at torsion elements, the root systems $R(m)$; in particular, they introduced a constant $d_m$ which is always the dimension of a representation of the semisimple, simply-connected group with root system dual to $R(m)$ and equals $1$ if the roots of height $m$ form a base of $R(m)$, and proved this property when $R$ is of type $A$ or $C$, and also in type $B$ if $m$ is odd. In this paper, we complete their analysis by determining a base of $R(m)$ and computing the constant $d_m$ in all cases.

Classification of the root systems $R(m)$

TL;DR

This work completes the analysis of root subsystems obtained from a reduced irreducible root system by selecting roots whose height is a multiple of (with Coxeter number and ). By combining a height-partition approach with detailed case analysis across all root-system types, it identifies a Levi-type base for in almost all cases and isolates precise exceptional configurations where an extra root appears and . The results unify and extend prior work of Nadimpalli–Pattanayak–Prasad by giving explicit , the associated weight (often a minuscule fundamental weight), and the Weyl-dimension-derived constant for every type, including the exceptional types and the low-rank cases . The findings sharpen the understanding of in the context of torsion-element character theory and connect the subsystem structure to representations of the dual root system via , with concrete tableaux and Dynkin-diagram constructions provided where needed. This classification aids precise computations of representation dimensions and clarifies when behaves as a standard Levi-type subsystem versus when extra nodes alter the Levi structure.

Abstract

Let be a reduced irreducible root system, its Coxeter number and a positive integer smaller than . Choose of base of , whence a corresponding height function, and let be the set of roots whose height is a multiple of . In a recent paper, S. Nadimpalli, S. Pattanayak and D. Prasad studied, for the purposes of character theory at torsion elements, the root systems ; in particular, they introduced a constant which is always the dimension of a representation of the semisimple, simply-connected group with root system dual to and equals if the roots of height form a base of , and proved this property when is of type or , and also in type if is odd. In this paper, we complete their analysis by determining a base of and computing the constant in all cases.

Paper Structure

This paper contains 9 sections, 9 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Statement of the result
  3. Proofs
  4. The partition of heights
  5. Notation
  6. Types $A_{n-1}$ and $C_n$
  7. Types $B_n$ and $D_{n+1}$
  8. Types $E_6, E_7, E_8$
  9. Types $F_4$ and $G_2$

Key Result

Theorem 2.1

(1) $R_m$ is always contained in a $W$-conjugate of $\Delta$. (2) One has $\Gamma(m) = R_m$ and hence $R(m)$ is of Levi type, in all cases except the cases below. In these cases, $\Gamma(m)$ is the union of $R_m$ and a root $\delta_{2m}$ of height $2m$, is not of Levi type and, denoting by $X^\dag$ (3) So, when $d_m\not=1$ it corresponds to the natural representation or a spin one of a simple fac

Theorems & Definitions (14)

  • Theorem 2.1
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • Lemma 3.4
  • proof
  • Corollary 3.6
  • proof
  • Lemma 3.7
  • ...and 4 more