turkmath.org

Let G and H be finite groups and let B and C be p-block algebras of G and H, respectively. In a landmark paper, Broué defined the notion of a perfect isometry I between B and C as a bijection between their irreducible characters with signs satisfying certain arithmetic conditions. He proved that the ratio of the codegrees of corresponding irreducible characters (including the sign) leads to a nonzero element β(I) in the field with p elements which is independent of the irreducible characters. We call β(I) the Broué invariant of I and show that if I comes from a p-permutation equivalence or a splendid Rickard equivalence between A and B then - up to a sign - it is independent of the equivalence and explicitly determined by local invariants of B and C.