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Stable perfect isometries between blocks of finite groups


Let $(\KK, \calO, F)$ be a large enough $p$-modular system for finite groups $G$ and $H$. Let $A$ be a $p$-block of the group algebra $\calO G$ and $B$ be a $p$-block of the group algebra $\calO H$. Brou {e} introduced the definition of a perfect isometry between the $p$-blocks $A$ and $B$ which is a generalized $\mathbb{K}$-valued character leading to a special bijection between the sets of irreducible $\KK$-characters of $A$ and $B$. We introduce and investigate the notion of a stable perfect isometry, a way to consider perfect isometries up to generalized projective characters of the corresponding $p$-blocks. The main interest lies in understanding for which blocks all stable perfect self-isometries can be lifted to perfect self-isometries. We verify this for algebras of abelian $p$-groups and certain cases of blocks with cyclic defect group as well as blocks with Klein four defect group. We also introduce the notion of a stable $p$-permutation equivalence. Given block $A$, we show that if all stable perfect self-isometries of $A$ lift to perfect self-isometries, then all stable $p$-permutation self-equivalences of $A$ lift to $p$-permutation self-equivalences.

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