Speed’s Orbit Problem: Variations on Themes of Burnside

✍ Scribed by Jonathan D.H. Smith

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 117 KB
• Volume: 20
• Category: Article
• ISSN: 0195-6698
• DOI: 10.1006/eujc.1999.0303

✦ Synopsis

The paper offers a solution to T. P. Speed's Orbit Problem, describing the orbit decomposition of direct powers of a transitive finite group action. The solution is given in terms of the Burnside algebra of the group and the right action of an incidence algebra of the subgroup lattice on a module of finitary functions. As a corollary, it is shown that almost all orbits in powers of a faithful action are regular. This gives one possible 'GF(1)' version of Burnside's original theorem that each irreducible complex representation of a finite group appears in a tensor power of a fixed faithful representation.