Description:

Title: The image of the power operations on Burnside rings

Abstract: The Burnside ring A(G) of a finite group G is the archetypal ring in G-equivariant homotopy theory, appearing as the zeroth stable homotopy of the G-equivariant sphere spectrum. As part of the structure of this homotopy data, Burnside rings admit power operations sending a finite G-set X to the set Xⁿ with compatible actions of both G and a symmetric group Sₙ. We study the smallest subring of A(G≀Sₙ) generated by a subset of the canonical basis and containing the image of these power operations. We provide three equivalent characterizations of this ring: (1) by universal property; (2) by generators and relations; and (3) by means of a character theory / table of marks. Using this character theory, we give formulas for the power operations in the table of marks.