Brunat, Olivier; Gramain, Jean-Baptiste Perfect isometries between blocks of complex reflection groups. (English) Zbl 1455.20012 J. Algebra 558, 260-292 (2020). In the paper under review, the authors prove the following important result. Given any integers \(d\), \(e\), \(r\) and \(r'\), and a prime \(p\) not dividing \(d\cdot e\), any two blocks of the complex reflection groups \(G(de, e, r)\) and \(G(de, e, r')\) with the same \(p\)-weight are perfectly isometric. Reviewer: Enrico Jabara (Venezia) MSC: 20C30 Representations of finite symmetric groups 20C15 Ordinary representations and characters 20C20 Modular representations and characters Keywords:perfect isometries; complex reflection groups PDFBibTeX XMLCite \textit{O. Brunat} and \textit{J.-B. Gramain}, J. Algebra 558, 260--292 (2020; Zbl 1455.20012) Full Text: DOI arXiv Link References: [1] Broué, M., Isométries parfaites, types de blocs, catégories dérivées, Astérisque, 181-182, 61-92 (1990) · Zbl 0704.20010 [2] Brunat, O.; Gramain, J.-B., Perfect isometries and Murnaghan-Nakayama rules, Trans. Amer. Math. Soc., 369, 11, 7657-7718 (2017) · Zbl 1476.20015 [3] Enguehard, M., Isométries parfaites entre blocs de groupes symétriques, Astérisque, 181-182, 157-171 (1990) · Zbl 0745.20012 [4] James, G.; Kerber, A., The Representation Theory of the Symmetric Group, Encyclopedia of Mathematics and Its Applications, vol. 16 (1981), Addison-Wesley Publishing Co.: Addison-Wesley Publishing Co. Reading, MA · Zbl 0491.20010 [5] Marin, I.; Michel, J., Automorphisms of complex reflection groups, Represent. Theory, 14, 747-788 (2010) · Zbl 1220.20036 [6] Morris, A. O.; Olsson, J. B., On p-quotients for spin characters, J. Algebra, 119, 1, 51-82 (1988) · Zbl 0686.20009 [7] Navarro, G., Characters and Blocks of Finite Groups, London Mathematical Society Lecture Note Series, vol. 250 (1998), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0903.20004 [8] Osima, M., On the representations of the generalized symmetric group. II, Math. J. Okayama Univ., 6, 81-97 (1956) · Zbl 0072.26003 [9] Pfeiffer, G., Character tables of Weyl groups in GAP, Bayreuth. Math. Schr., 47, 165-222 (1994) · Zbl 0830.20023 [10] Read, E. W., On the finite imprimitive unitary reflection groups, J. Algebra, 45, 2, 439-452 (1977) · Zbl 0348.20003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.