## Journal of the Mathematical Society of Japan

### On Alperin's weight conjecture for $p$-blocks of $p$-solvable groups

Masafumi MURAI

#### Abstract

For $p$-solvable groups, a strong form of Alperin's weight conjecture has been proved by T. Okuyama (unpublished). L. Barker has refined this theorem by taking Green correspondence into account. We prove here a relative version of Barker's theorem.

#### Article information

Source
J. Math. Soc. Japan, Volume 65, Number 4 (2013), 1037-1054.

Dates
First available in Project Euclid: 24 October 2013

https://projecteuclid.org/euclid.jmsj/1382620184

Digital Object Identifier
doi:10.2969/jmsj/06541037

Mathematical Reviews number (MathSciNet)
MR3127815

Zentralblatt MATH identifier
1290.20009

Subjects
Primary: 20C20: Modular representations and characters

#### Citation

MURAI, Masafumi. On Alperin's weight conjecture for $p$-blocks of $p$-solvable groups. J. Math. Soc. Japan 65 (2013), no. 4, 1037--1054. doi:10.2969/jmsj/06541037. https://projecteuclid.org/euclid.jmsj/1382620184

#### References

• J. L. Alperin and M. Broué, Local methods in block theory, Ann. of Math. (2), 110 (1979), 143–157.
• L. Barker, On $p$-soluble groups and the number of simple modules associated with a given Brauer pair, Quart. J. Math. Oxford Ser. (2), 48 (1997), 133–160.
• D. W. Burry, A strengthened theory of vertices and sources, J. Algebra, 59 (1979), 330–344.
• E. C. Dade, Endo-permutation modules over $p$-groups. I, Ann. of Math. (2), 107 (1978), 459–494.
• W. Feit, The Representation Theory of Finite Groups, North-Holland Math. Library, 25, North-Holland, Amsterdam, 1982.
• B. Huppert and N. Blackburn, Finite Groups. II, Grundlehren Math. Wiss., 242, Springer-Verlag, Berlin, 1982.
• B. Huppert and N. Blackburn, Finite Groups. III, Grundlehren Math. Wiss., 243, Springer-Verlag, Berlin, 1982.
• I. M. Isaacs, Character Theory of Finite Groups, Pure Appl. Math. (Amst.), 69, Academic Press, New York, 1976.
• I. M. Isaacs and G. Navarro, Weights and vertices for characters of $\pi$-separable groups, J. Algebra, 177 (1995), 339–366.
• A. Laradji, On normal subgroups and simple modules with a given vertex in a $p$-solvable group, J. Algebra, 308 (2007), 484–492.
• M. Murai, Normal subgroups and heights of characters, J. Math. Kyoto Univ., 36 (1996), 31–43.
• H. Nagao and Y. Tsushima, Representations of Finite Groups, Academic Press, New York, 1989.
• L. Puig, Local block theory in $p$-solvable groups, In: The Santa Cruz Conference on Finite Groups, Univ. California, Santa Cruz, Calif., 1979, (Eds. B. Cooperstein and G. Mason), Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, RI, 1980, pp.,385–388.
• J. Thévenaz, $G$-algebras and Modular Representation Theory, Oxford Math. Monogr., Clarendon Press, Oxford, 1995.
• A. Watanabe, Normal subgroups and multiplicities of indecomposable modules, Osaka J. Math., 33 (1996), 629–635.