Navarro, Gabriel; Tiep, Pham Huu Brauer’s height zero conjecture for the 2-blocks of maximal defect. (English) Zbl 1280.20011 J. Reine Angew. Math. 669, 225-247 (2012). Let \(G\) be a finite group, \(p\) be a prime integer and let \(B\) be a \(p\)-block of \(G\) with defect group \(D\). The Height Zero Conjecture of R. Brauer: “All irreducible complex characters in \(B\) have height zero if and only if \(D\) is Abelian” has stimulated a lot of research in Finite Group Modular Representation Theory since it conjectures a relationship between the “structure” of \(B\) and the degrees of the irreducible complex characters in \(B\). Utilizing the Classification of Finite Simple Groups and many results in Block Theory, etc., the authors prove: Theorem A. Let \(B\) be a 2-block of the finite group \(G\) with a defect group \(P\in\text{Syl}_2(G)\). Then \(\chi(1)\) is odd for all \(\chi\in\text{Irr}(B)\) if and only if \(P\) is Abelian. In Section 7, the authors present a possible program for establishing Theorem A for odd primes \(p\). Reviewer: Morten E. Harris (Chicago) Cited in 23 Documents MSC: 20C20 Modular representations and characters 20C15 Ordinary representations and characters Keywords:finite groups; \(p\)-blocks; defect groups; height zero conjecture; Brauer conjectures; degrees of irreducible complex characters PDFBibTeX XMLCite \textit{G. Navarro} and \textit{P. H. Tiep}, J. Reine Angew. Math. 669, 225--247 (2012; Zbl 1280.20011) Full Text: DOI References: [1] DOI: 10.1016/0021-8693(76)90135-6 · Zbl 0367.20006 · doi:10.1016/0021-8693(76)90135-6 [2] Berger T., Nagoya Math. J. 109 pp 109– (1988) · Zbl 0637.20006 · doi:10.1017/S0027763000002798 [3] DOI: 10.1016/j.jpaa.2006.01.013 · Zbl 1113.20012 · doi:10.1016/j.jpaa.2006.01.013 [4] DOI: 10.1006/jabr.1998.7661 · Zbl 0930.20008 · doi:10.1006/jabr.1998.7661 [5] Bonnafé C., Publ. Math. Inst. Hautes E’t. Sci. 97 pp 1– (2003) · Zbl 1054.20024 · doi:10.1007/s10240-003-0013-3 [6] Broué M., Astérisque 181 pp 61– (1990) [7] Broué M., Math. 395 pp 56– (1989) [8] DOI: 10.1016/0021-8693(76)90088-0 · Zbl 0334.20008 · doi:10.1016/0021-8693(76)90088-0 [9] DOI: 10.1016/0021-8693(77)90371-4 · Zbl 0376.20003 · doi:10.1016/0021-8693(77)90371-4 [10] DOI: 10.1016/j.jalgebra.2007.06.036 · Zbl 1194.20048 · doi:10.1016/j.jalgebra.2007.06.036 [11] Trans. Amer. Math. Soc. 98 pp 263– (1961) [12] DOI: 10.1007/BF01232665 · Zbl 0811.20009 · doi:10.1007/BF01232665 [13] DOI: 10.1080/00927879008823932 · Zbl 0696.20011 · doi:10.1080/00927879008823932 [14] Gluck D., Illinois J. Math. 27 pp 514– (1983) [15] DOI: 10.2307/1999582 · Zbl 0543.20007 · doi:10.2307/1999582 [16] DOI: 10.1016/0021-8693(84)90168-6 · Zbl 0534.20003 · doi:10.1016/0021-8693(84)90168-6 [17] DOI: 10.1112/jlms/s2-41.1.63 · Zbl 0661.20030 · doi:10.1112/jlms/s2-41.1.63 [18] DOI: 10.1007/BF01109967 · Zbl 0198.04502 · doi:10.1007/BF01109967 [19] DOI: 10.1016/0021-8693(84)90058-9 · Zbl 0526.20006 · doi:10.1016/0021-8693(84)90058-9 [20] DOI: 10.1007/s00222-007-0057-y · Zbl 1138.20010 · doi:10.1007/s00222-007-0057-y [21] DOI: 10.1006/jabr.1995.1014 · Zbl 0840.20006 · doi:10.1006/jabr.1995.1014 [22] DOI: 10.1023/A:1025923522954 · doi:10.1023/A:1025923522954 [23] DOI: 10.1016/S0021-8693(03)00229-1 · Zbl 1045.20010 · doi:10.1016/S0021-8693(03)00229-1 [24] DOI: 10.4171/JEMS/220 · Zbl 1205.20011 · doi:10.4171/JEMS/220 [25] DOI: 10.1081/AGB-100002175 · Zbl 1004.20003 · doi:10.1081/AGB-100002175 [26] DOI: 10.1016/j.aim.2004.11.008 · Zbl 1097.20012 · doi:10.1016/j.aim.2004.11.008 [27] Murai M., J. Math. Kyoto Univ. 35 pp 607– (1995) [28] DOI: 10.4007/annals.2004.160.1129 · Zbl 1079.20010 · doi:10.4007/annals.2004.160.1129 [29] DOI: 10.1016/j.jalgebra.2005.02.005 · Zbl 1087.20013 · doi:10.1016/j.jalgebra.2005.02.005 [30] J., Ann. Math. 89 (2) pp 405– (1969) [31] DOI: 10.2307/1994333 · Zbl 0139.24902 · doi:10.2307/1994333 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.