On the ideals and essential algebras of shifted functors of linear representations

Abstract

We present a study on the shifted Green biset functors kR F,G of linear F-representations with coefficients over k, for fields k and F of characteristic zero and a finite group G. We provide a criterion for the vanishing of their essential algebras and we prove that the condition of uniqueness of minimal groups for simple modules holds for these functors. We give a parametrization of a family of simple kR Q,G -modules. We also prove the semisimplicity of the category of CR C,G -modules by means of an equivalence with a category of modules over a semisimple algebra.

Green fields

“…It is worth saying that the fact that CR C satisfies condition 1 of Corollary 4.2 was first shown in Proposition 4.3 of [12]. Also, the equivalence by evaluation at 1 between C(R C ) K -Mod and CR C (K)-Mod was first given in Proposition 4.3 of [10].…”

Essential support of Green biset functors via morphisms

“…-biset for the actions (k 1 , h 1 , g 1 ) • (k, g, h, g 0 ) • (k 2 , g 2 , h 2 , g 3 ) = (k 1 kk 2 , g 1 gg 2 , h 1 hh 2 , g 1 g 0 g 3 ), and with identity element AG = A(Inf 1×G 1 )( A ). As a consequence of the Yoneda lemma, the functor A G is a projective A-module, and as G runs over a set of representatives of the isomorphism classes of D, we obtain a set of projective generators of A − Mod (see G. [6,Subsection 2.3]). As a Green biset functor, little is known on their modules…”

“…Let k and F be fields of characteristic zero. We know from G. [6,Proposition 3.21] that Supp(kR F,G ) consists only of cyclic groups, so the isomorphism classes of simple modules are in correspondence with the isomorphism classes of seeds. Since κ G is an ideal of kR F,G , from G. [6, Theorem 3.12], we know that κ G = I E for some E ⊂ c k∩F (G), where…”

“…] is indeed a bijection. Some well-known Green biset functors satisfy such condition, e.g., kB [3], kR F for fields k and F of characteristic zero and its shifted functors ( [1,6]), and kB Cp for a prime number p [7].…”

Essential support of Green biset functors via morphisms

“…Romero proved in [10,Proposition 4.2] that if A satisfies that any two minimal groups for a simple A-module are isomorphic, then the correspondence [(H, V )] → [S H,V ] is actually a bijection between the sets of isomorphism classes. Some well-known Green biset functors satisfy this uniqueness condition, e.g., the Burnside functor kB [3], the functor of linear representations kR F for fields k and F of characteristic zero and its shifted functors ( [1], [8]), and the fibered Burnside functor kB Cp for a prime number p [9]. In this case, a better understanding of the correspondence between seeds and simple modules requires to go further in the study of the essential algebras and their simple modules.…”