arXiv subjects
Publications and source records attributed to Benjamín García.
In this paper, we describe the indecomposable factors of a fibered Burnside ring in terms of bases coming from the standard basis. We provide a further characterization of each factor as the solvable component of the fibered Burnside ring of a corresponding Weyl group.
Fibered Burnside rings appear as Grothendieck rings of fibered permutation representations of a finite group, generalizing Burnside rings and monomial representation rings. Their species, primitive idempotents and their conductors are of particular interest in representation theory as they encode information related to the structure of the group. In this note, we introduce fiber change maps between fibered Burnside rings, and we present results on their functoriality and naturality with respect to biset operations. We present some advances on the conductors for cyclic fiber groups, and fully determine them in particular cases, covering a wide range of interesting examples.
Let $G$ and $H$ be finite groups. We give a condition on $G$ and $H$ that implies that the $A$-fibered Burnside rings $B^A(G)$ and $B^A(H)$ are isomorphic. As a consequence, we show the existence of non-isomorphic groups $G$ and $H$ such that $B^A(G)$ and $B^A(H)$ are isomorphic rings. Here, the abelian fiber group $A$ can be chosen in a non-trivial way, that is, such that $B^A(G)$ and $B^A(H)$ are strictly bigger than the Burnside rings of $G$ and $H$, for which such counterexamples are already known.
In this article, we propose a concept of anti-involution for Green biset functors and we provide equivalent definitions. We present $\star$-Green biset functors and we study their orthogonal units and the orthogonal automorphisms in their associated categories.
In this note, we present a notion of species isomorphism for fibered Burnside rings. We prove that a species isomorphism can be restricted to an isomorphism of mark tables, that it can be extended to an isomorphism between the ghost rings and we provide three additional characterizations of this concept.
We present a very natural but yet useful criterion to detect vanishing of essential algebras of a Green biset functor $A$ by means of morphisms. We introduce the morphisms $Inf:A \rightarrow A_G$ and $Res:A_G\rightarrow A$ to prove that the class of groups for which the essential algebras do not vanish is the same for $A$ and any shifted functor $A_G$. We use the kernel of $Res$ to give a characterization of the seeds of simple $A_G$-modules which are not obtained as pullback of simple $A$-modules.
We present a study on the Yoneda-Dress construction of biset functors of linear representations over a field of characteristic zero. We give a characterization of their lattices of ideals and we provide a criterion of vanishing for their essential algebras. We provide a parametrization for a family of simple modules over the functor of rational representations. Finally, we give an equivalence between the category of modules over the shifted functor of complex class functions and the category of modules over a semisimple algebra.