Perfect isometries and isotypies for blocks with Abelian defect groups and the inertial quotients isomorphic to \(\mathbf Z_ 3\times\mathbf Z_ 3\)
Let \(p\) be a prime number, let \(\mathcal O\) be a complete discrete valuation ring with algebraically closed residue field \(k\) of characteristic \(p\) and with quotient field \(K\) of characteristic zero. Let \(G\) be a finite group and let \(b\) be a block of \({\mathcal O}G\) with defect group \(P\). Also let \(e\) be a root of \(b\) in \(C_G(P)\) (i.e., a block of \({\mathcal O}C_G(P)\) such that \(e^G=b\)). The inertial quotient of \(b\) is the group \(E=N_G(P,e)/(PC_G(P))\). Assume also that \(\mathcal O\) contains all \(|P|\)-th roots of unity. The main result of this paper is: Theorem 1.12. Assume in the above that \(P\) is abelian, that \(p\neq 2,7\) and that \(E\cong Z_3\times Z_3\). Then \(b\), its Brauer correspondent \(\text{Br}_P(b)\) and \(e\) considered as blocks of \({\mathcal O}G\), \({\mathcal O}N_G(P)\) and \({\mathcal O}N_G(P,e)\) resp. are all isotypic (in the ``good definition'' of an isotopy). (See Definitions 1.7 and 1.8 for the definition of an isotypy and the ``good definition'' of an isotypy).
scientific article; zbMATH DE number 917729