On the image of the trivial source ring in the ring of virtual characters of a finite group

Abstract

We introduce a new type of equivalence between blocks of finite group algebras called a strong isotypy. A strong isotypy is equivalent to a p-permutation equivalence and restricts to an isotypy in the sense of Broué. To prove these results we first establish that the group T O (B) of trivial source B-modules, where B is a block of a finite group algebra, is isomorphic to groups of "coherent character tuples." This provides a refinement of work by Boltje and Carman which characterizes the ring T O (G) of trivial source OG-modules, where G is a finite group, in terms of coherent character tuples.