Perfect isometries and the Isaacs correspondence.
The author shows that the Isaacs character correspondence \(^*\colon\text{Irr}_{q'}(G)\to\text{Irr}_{q'}(N_G(Q))\) induces a perfect isometry between certain \(p\)-blocks of \(G\) and \(N_G(Q)\), thus complementing similar results of \textit{A.~Watanabe} [J. Algebra 216, No. 2, 548-565 (1999; Zbl 0936.20006)] and \textit{H.~Horimoto} [Hokkaido Math. J. 30, No. 1, 65-74 (2001; Zbl 0992.20006)] on the Glauberman correspondence. The main result reads as follows. Let \(G\) be a finite group of odd order, \(p\) and \(q\) distinct primes, \(B\) a \(p\)-block of \(G\) such that every irreducible character of \(B\) has \(q'\)-degree, and let \(D\) be a defect group of \(B\). Then there is \(Q\in\text{Syl}_q(G)\) and a unique \(p\)-block \(B^*\) of \(N_G(Q)\) with defect group \(D\) such that \(\text{Irr}(B^*)=\{\chi^*\mid\chi\in\text{Irr}(B)\}\). Moreover, there is a perfect isometry \(\chi\mapsto\chi^*\), \(\chi\in\text{Irr}(B)\).
scientific article; zbMATH DE number 2002259