The slice Burnside ring and the section Burnside ring of a finite group
The Burnside ring \(B(G)\) of a finite group \(G\) is the Grothendieck ring of the category of finite \(G\)-sets. In the paper under review, the author introduces two variants of this construction, the slice Burnside ring \(\Xi(G)\) and the section Burnside ring \(\Gamma(G)\). Here \(\Xi(G)\) is the Grothendieck ring of the category of all morphisms between finite \(G\)-sets, and \(\Gamma(G)\) is the Grothendieck ring of the category of all Galois morphisms between finite \(G\)-sets where a morphism of finite \(G\)-sets \(f:X \longrightarrow Y\) is called a Galois morphism if \(G_x = G_{x'}\) for all \(x,x' \in G\) such that \(f(x) = f(x')\).NEWLINENEWLINEThe author develops a structure theory for \(\Xi(G)\) and \(\Gamma(G)\) which parallels the corresponding theory for \(B(G)\). Both are commutative rings, and free of finite rank as abelian groups; moreover, \(\Gamma(G)\) is a subring of \(\Xi(G)\). Both \(\Xi(G)\) and \(\Gamma(G)\) can be embedded into ghost rings, and the images of these embeddings can be characterized via certain congruences. Moreover, there are ring homomorphisms \(s_G:\Xi(G) \longrightarrow B(G)\) and \(i_G:B(G) \longrightarrow \Xi(G)\) such that the image of \(i_G\) is contained in \(\Gamma(G)\) and \(s_G \circ i_G = \text{id}_{B(G)}\). Tensoring with \(\mathbb{Q}\) leads to split semisimple \(\mathbb{Q}\)-algebras \(\mathbb{Q} \Xi (G)\) and \(\mathbb{Q} \Gamma (G)\). Explicit formulas for the idempotents of \(\mathbb{Q} \Xi (G)\) and \(\mathbb{Q} \Gamma (G)\) can be given, and the prime spectra of \(\Xi(G)\) and \(\Gamma(G)\) can be determined. The correspondences \(G \longmapsto \Xi(G)\) and \(G \longmapsto \Gamma(G)\) define Green biset functors. The author also deals with the unit groups of \(\Xi(G)\) and \(\Gamma(G)\). They are elementary abelian finite \(2\)-groups. However, in contrast to the unit groups of \(B(G)\), they cannot be endowed with a biset functor structure.
scientific article; zbMATH DE number 6051029
scientific article