UC Santa Cruz

A new type of equivalence between blocks of finite group algebras called a \textit{strong isotypy} is introduced. We show that a strong isotypy restricts to an isotypy in the sense of Brou {e}. We also show that a strong isotypy determines, and is determined by, a $p$-permutation equivalence. To prove these results we first construct isomorphisms between the group $T_{\OO}(B)$ of trivial source $B$-modules, where $B$ is a block of a finite group algebra, and groups of ``coherent character tuples.'' This provides a refinement of work by Boltje and Carman which characterizes the ring $T_{\OO}(G)$ of trivial source $\OO G$-modules, where $G$ is a finite group, in terms of coherent character tuples.