Groups of perfect self-isometries

Let [math]B[/math] be a block of [math]\mathcal{O}G[/math] for a finite group [math]G[/math]. The set of perfect isometries from [math]B[/math] to itself forms a group [math]{\rm PI}(B)[/math] under composition, and its isomorphism class is a derived invariant. Write [math]{\rm PI}_+(B)[/math] for the subgroup of [math]{\rm PI}(B)[/math] consisting of isometries with all signs positive (this is different to the subgroup [math]{\rm PI}^+(B)[/math] defined in [Ru11]). If [math]{\rm Piccent}(B)=1[/math], then [math]{\rm Pic}(B)[/math] embeds into [math]{\rm PI}_+(B)[/math].

The group [math]{\rm PI}(B)[/math] is useful in the calculation of Picard groups (see [EL18c]) and in the calculation of extensions of Morita equivalence classes from normal subgroups of index [math]p[/math] (see for example [Wa00] and [EL18a]).