Solutions of Reeder's Puzzle

TL;DR

This paper provides a comprehensive solution to the generalized Reeder's puzzle on Dynkin and affine Dynkin diagrams, determining the number of equivalence classes and extending results to certain trees containing E_6.

Contribution

It offers a detailed classification and solution for the puzzle on complex graph structures, including new results on simply-laced trees with E_6 subgraphs.

Findings

Number of equivalence classes for Dynkin diagrams

Number of equivalence classes for affine Dynkin diagrams

Main theorem on simply-laced trees containing E_6

Abstract

In this paper we consider the generalized Reeder's puzzle, introduced by Reeder in 2005 and generalized by Borovoi and Evenor in 2016. We give a detailed solution of the puzzle for the graphs of Dynkin diagrams and affine Dynkin diagrams. We find the number of equivalence classes in each case. We also discuss more general graphs, and prove the main theorem about graphs (simply-laced trees) that contain E_6 as a subgraph.