## Real characters in blocks.(English)Zbl 1264.20012

This paper proposes the study of “real versions” of some open problems in the modular representation theory of finite groups. These conjectures are first stated and then proved under some more special hypotheses. Real versions are presented of Brauer’s $$k(B)$$-conjecture, Olsson’s conjecture and a conjecture of Eaton. For example, the “strong real version of Brauer’s $$k(B)$$-conjecture” reads as follows: Let $$B$$ be a 2-block of the finite group $$G$$ with defect group $$D$$. Then the number of real valued characters in $$B$$ is bounded above by the number of those elements in $$D$$ for which there exists an element in $$N_G(D)$$ that conjugates it to its inverse. In the “weak real version” the $$N_G(D)$$ is replaced by $$G$$ in the above statement. The strong version is proved for symmetric groups and also when $$D$$ is central, and also under some other conditions on $$D$$. The last part of the paper deals with fusion of elements of defect groups.

### MSC:

 20C20 Modular representations and characters 20C15 Ordinary representations and characters

GAP
Full Text:

### References:

 [1] R. Brauer: Some applications of the theory of blocks of characters of finite groups , IV, J. Algebra 17 (1971), 489-521. · Zbl 0247.20013 [2] R. Brauer: Representations of finite groups ; in Lectures on Modern Mathematics 1 , Wiley, New York, 133-175, 1963. · Zbl 0124.26504 [3] R. Brauer: Number theoretical investigations on groups of finite order ; in Proceedings of the International Symposium on Algebraic Number Theory, Tokyo and Nikko, 1955, Science Council of Japan, Tokyo, 55-62, 1956. · Zbl 0073.01403 [4] C.W. Curtis and I. Reiner: Methods of Representation Theory, II, Wiley, New York, 1987. [5] S. Dolfi, G. Navarro and P.H. Tiep: Primes dividing the degrees of the real characters , Math. Z. 259 (2008), 755-774. · Zbl 1149.20006 [6] L. Dornhoff: Group Representation Theory, Part B, Dekker, New York, 1972. · Zbl 0236.20004 [7] The GAP Group: GAP–Groups, Algorithms and Programming, Version 4.4.12, http://www.gap-system.org, 2008. [8] R. Gow: Real $$2$$-blocks of characters of finite groups , Osaka J. Math. 25 (1988), 135-147. · Zbl 0656.20015 [9] R. Gow and J. Murray: Real 2-regular classes and 2-blocks , J. Algebra 230 (2000), 455-473. · Zbl 0963.20004 [10] C.W. Eaton: Generalisations of conjectures of Brauer and Olsson , Arch. Math. (Basel) 81 (2003), 621-626. · Zbl 1046.20007 [11] L. Héthelyi and E. Horváth: Galois actions on blocks and classes of finite groups , J. Algebra 320 (2008), 660-679. · Zbl 1156.20009 [12] I.M. Isaacs: Character Theory of Finite Groups, Academic Press, New York, 1976. [13] I.M. Isaacs, G. Malle and G. Navarro: Real characters of $$p'$$-degree , J. Algebra 278 (2004), 611-620. · Zbl 1080.20007 [14] I.M. Isaacs and G. Navarro: Group elements and fields of character values , J. Group Theory 12 (2009), 635-650. · Zbl 1186.20009 [15] S. Iwasaki: On finite groups with exactly two real conjugate classes , Arch. Math. (Basel) 33 (1979), 512-517. · Zbl 0433.20014 [16] G. James and A. Kerber: The Representation Theory of the Symmetric Group, Addison-Wesley Publishing Co., Reading, MA, 1981. · Zbl 0491.20010 [17] S.G. Kolesnikov and Ja. N. Nuzhin: On strong reality of finite simple groups , Acta Appl. Math. 85 (2005), 195-203. · Zbl 1086.20010 [18] P. Landrock: Finite Group Algebras and Their Modules, London Mathematical Society Lecture Note Series 84 , Cambridge Univ. Press, Cambridge, 1983. · Zbl 0523.20001 [19] A. Moretó and G. Navarro: Groups with three real valued irreducible characters , Israel J. Math. 163 (2008), 85-92. · Zbl 1152.20011 [20] J. Murray: Involutions, extended defect groups of real $$2$$-blocks and vertex theory , [21] J. Murray: Real subpairs and Frobenius-Schur indicators of characters in 2-blocks , J. Algebra 322 (2009), 489-513. · Zbl 1193.20007 [22] H. Nagao and Y. Tsushima: Representations of Finite Groups, Academic Press, Boston, MA, 1989. · Zbl 0673.20002 [23] G. Navarro: Characters and Blocks of Finite Groups, London Mathematical Society Lecture Note Series 250 , Cambridge Univ. Press, Cambridge, 1998. · Zbl 0903.20004 [24] G. Navarro, L. Sanus and P.H. Tiep: Groups with two real Brauer characters , J. Algebra 307 (2007), 891-898. · Zbl 1124.20006 [25] G. Navarro, L. Sanus and P.H. Tiep: Real characters and degrees , Israel J. Math., · Zbl 1186.20010 [26] J.B. Olsson: McKay numbers and heights of characters , Math. Scand. 38 (1976), 25-42. · Zbl 0327.20005 [27] J.B. Olsson: Inequalities for block-theoretic invariants ; in Representations of Algebras (Puebla, 1980), Lecture Notes in Math. 903 , Springer, Berlin, 270-284, 1981. · Zbl 0478.20008 [28] J.B. Olsson: On the number of characters in blocks of finite general linear, unitary and symmetric groups , Math. Z. 186 (1984), 41-47. · Zbl 0534.20006
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.