Charles Eaton: Created page with "Let $B$ be a block of $\mathcal{O}G$ for a finite group $G$ . The set of perfect isometries from $B$ to itself..."

2018-12-06T09:40:20Z

Created page with "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..."

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Let <math>B</math> be a block of <math>\mathcal{O}G</math> for a finite group <math>G</math>. The set of [[Perfect isometry|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 [[References|[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 [[References|[EL18c]]]) and in the calculation of extensions of Morita equivalence classes from normal subgroups of index <math>p</math> (see for example [[References|[Wa00]]] and [[References|[EL18a]]]).

Groups of perfect self-isometries - Revision history

Charles Eaton: Created page with "Let B be a block of \mathcal{O}G for a finite group G. The set of perfect isometries from B to itself..."

Charles Eaton: Created page with "Let $B$ be a block of $\mathcal{O}G$ for a finite group $G$ . The set of perfect isometries from $B$ to itself..."