Character triples and equivalences over a group graded G-algebra

Group graded Morita equivalences for wreath products

“…In this article, we continue the study done in [2], [3] and [4], and we obtain group graded Morita equivalences for tensor products (Proposition 3.3) and wreath products (Theorem 5.3). The main motivation for such constructions in the representation theory of finite groups is given by the fact that in order to prove most reduction theorems, recent results of Britta Späth, surveyed in [5], [6] and [7], show that a new character triple can be constructed via a wreath product construction of character triples ([7, Theorem 2.21]).…”

“…Note that most results in this paper will utilize "Ḡ-gradings", although this is not essential: one may consider instead the gradings to be given directly by G. The reasoning behind this particular choice is to match our notations previously used in articles [2] and [3], given that our main application for the results of this project is the stronglyḠ-graded algebra A = bOG, where b is aḠ-invariant block of ON .…”

“…The main motivation for such constructions in the representation theory of finite groups is given by the fact that in order to prove most reduction theorems, recent results of Britta Späth, surveyed in [5], [6] and [7], show that a new character triple can be constructed via a wreath product construction of character triples ([7, Theorem 2.21]). There is a link between character triples and group graded Morita equivalences, presented in [3], so we want to prove that a similar wreath product construction can also be made for the corresponding group graded Morita equivalences.…”

“…More precisely, in Theorem 6.7 of [3], it is proved that certain character triples relations utilized by Britta Späth: the first-order relation ( [7,Definition 2.1]) and the central-order relation ([7, Definition 2.7]), are consequences of a special type of group graded Morita equivalences induced by a graded bimodule over a G-graded G-acted algebra (usually denoted by C as in Section 2.3), where G is a finite group. More details about group graded Morita theory over C can be found in [4].…”

“…Another motivation comes from the fact that it is already known by [1,Theorem 5.1.21] that Morita equivalences can be extended to wreath products. This paper is organized as follows: In Section 2, we introduce the general notations and we recall from [3] the definitions of a G-graded G-acted algebra, of a G-graded algebra over C, of a G-graded bimodule over C and the notion of a Ggraded Morita equivalence over C. In Section 3, we prove that the previously recalled algebraic constructions are compatible with tensor products and the main proposition in this section, Proposition 3.3, proves that the tensor products of some group graded Morita equivalent algebras over some group graded group acted algebras remain group graded Morita equivalent over a group graded group acted algebra. In Section 4, we prove that the previously enumerated algebra types are also compatible with wreath products.…”