The Isaacs-Navarro conjecture for symmetric groups.
Let \(p\) be a prime, let \(B\) be a \(p\)-block of a finite group \(G\) with defect group \(D\), and let \(b\) be the Brauer correspondent of \(B\), a \(p\)-block of \(N_G(D)\) with defect group \(D\). The Alperin-McKay conjecture asserts that \(|M(B)|=|M(b)|\) where \(M(B)\) is the set of irreducible characters \(\chi\) of height 0 in \(B\). (This means that the \(p\)-part of \(|G|/\chi(1)\) is \(|D|\).) The Isaacs-Navarro conjecture [\textit{I. M. Isaacs, G. Navarro}, Ann. Math. (2) 156, No. 1, 333-344 (2002; Zbl 1018.20007)] is a recent refinement of the Alperin-McKay conjecture. It asserts that \(|M_k(B)|=|M_k(b)|\) for \(k=1,\dots,p\) where \[ M_k(B)=\{\chi\in M(B):\tfrac{|G:D|}{\chi(1)}\equiv\pm k\pmod p\} \] for each \(k\). In the paper under review, the author verifies this conjecture, for \(G\) a finite symmetric group, by constructing an explicit bijection between \(M_k(B)\) and \(M_k(b)\). His method refines work of \textit{J. B. Olsson} [in Math. Scand. 38, 25-42 (1976; Zbl 0327.20005)].
scientific article; zbMATH DE number 1901821