RT Conference Proceedings
T1 Butterfly, Möbius, and Double Burnside algebras of noncyclic finite groups
A1 Park, Sejong
K1 Grupos finitos
AB The double Burnside ring B(G,G) of a finite group G is theGrothendieck ring of finite (G,G)-bisets with respect to the tensorproduct of bisets over G. Many invariants of the group G, such as the(single) Burnside ring B(G) and the character ring R_C(G), are modulesover B(G,G). The double Burnside ring B(G,G) and its various subringsappear as crucial ingredients in functorial representation theory,homotopy theory, and the theory of fusion systems. It is known to besemisimple over rationals if and only if G is cyclic, and in this casean explicit isomorphism onto a direct product of full matrix algebras isgiven by Boltje and Danz (2013). But not much is know beyond that on theexplicit algebra structure.We generalize some techniques of Boltje and Danz for cyclic groups toarbitrary finite groups and as a result obtain an explicit isomorphismof the rational double Burnside algebra of a finite group G into a blocktriangular matrix algebra when all Sylow subgroups (for all primes) of Gare cyclic. Such groups can be characterized as groups G where theZassenhaus Butterfly lemma gives the meet of two sections (H, K) of Gwith respect to the subsection relation. Key ingredients are arefinement of the inclusion relation among subgroups of G x G and Möbiusinversion over various posets of subgroups.This is a joint work with Goetz Pfeiffer (Galway).
YR 2018
FD 2018-03-09
LK https://hdl.handle.net/10630/15360
UL https://hdl.handle.net/10630/15360
LA eng
NO Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech.
DS RIUMA. Repositorio Institucional de la Universidad de Málaga
RD 20 ene 2026