Coherence conditions for the characters of trivial source modules and strong isotypies

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 Brou\'{e}. To prove these results we first establish that the group $T_{\mathcal{O}}(B)$ 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 $T_{\mathcal{O}}(G)$ of trivial source $\mathcal{O}G$-modules, where $G$ is a finite group, in terms of coherent character tuples.