On exponentiation of G-sets

✍ Scribed by Andreas Blass

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 665 KB
• Volume: 135
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(93)e0109-h

✦ Synopsis

We show that Joyal's rule of signs in combinatorics arises naturally from Dress's concept of exponentiation of virtual G-sets. We also show that two finite G-sets admit a G-equivariant bijection between their power sets if and only if the (complex) linear representations they determine are equivalent.

1. Burnside rings and exponentiation

Let G be a finite group. By a G-set, we mean a finite set A equipped with a left action of G, which we write as g(u) or simply as ga. If A and B are two G-sets, then so are the disjoint union A + B with G acting on each summand separately, the Cartesian product A x B with G acting componentwise, and the set BA of functions from A to B with