Primitive spherical systems

TL;DR

This paper classifies primitive spherical systems, a key combinatorial structure in the theory of wonderful varieties, and establishes a correspondence between quotients of spherical systems and distinguished subsets of colors.

Contribution

It provides a complete classification of primitive spherical systems and links quotients of spherical systems to distinguished subsets of colors.

Findings

Complete classification of primitive spherical systems

Establishment of correspondence between quotients and distinguished subsets of colors

Application to the theory of wonderful varieties

Abstract

A spherical system is a combinatorial object, arising in the theory of wonderful varieties, defined in terms of a root system. All spherical systems can be obtained by means of some general combinatorial procedures (such as parabolic induction and wonderful fiber product) from the so-called primitive spherical systems. Here we classify primitive spherical systems. As an application, we prove that the quotients of a spherical system are in correspondence with the so-called distinguished subsets of colors.