Defect zero blocks for finite simple groups
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- by Andrew Granville and Ken Ono PDF
- Trans. Amer. Math. Soc. 348 (1996), 331-347 Request permission
Abstract:
We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a $p$-block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero $p-$blocks remained unclassified were the alternating groups $A_{n}$. Here we show that these all have a $p$-block with defect 0 for every prime $p\geq 5$. This follows from proving the same result for every symmetric group $S_{n}$, which in turn follows as a consequence of the $t$-core partition conjecture, that every non-negative integer possesses at least one $t$-core partition, for any $t\geq 4$. For $t\geq 17$, we reduce this problem to Lagrange’s Theorem that every non-negative integer can be written as the sum of four squares. The only case with $t<17$, that was not covered in previous work, was the case $t=13$. This we prove with a very different argument, by interpreting the generating function for $t$-core partitions in terms of modular forms, and then controlling the size of the coefficients using Deligne’s Theorem (née the Weil Conjectures). We also consider congruences for the number of $p$-blocks of $S_{n}$, proving a conjecture of Garvan, that establishes certain multiplicative congruences when $5\leq p \leq 23$. By using a result of Serre concerning the divisibility of coefficients of modular forms, we show that for any given prime $p$ and positive integer $m$, the number of $p-$blocks with defect 0 in $S_n$ is a multiple of $m$ for almost all $n$. We also establish that any given prime $p$ divides the number of $p-$modularly irreducible representations of $S_{n}$, for almost all $n$.References
-
G. Almkvist, private communication.
- George E. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. MR 0557013
- Richard Brauer, Representations of finite groups, Lectures on Modern Mathematics, Vol. I, Wiley, New York, 1963, pp. 133–175. MR 0178056 K. Erdmann and G. Michler, Blocks for symmetric groups and their covering groups and quadratic forms, preprint.
- Paul Fong and Bhama Srinivasan, The blocks of finite classical groups, J. Reine Angew. Math. 396 (1989), 122–191. MR 988550
- Frank Garvan, Dongsu Kim, and Dennis Stanton, Cranks and $t$-cores, Invent. Math. 101 (1990), no. 1, 1–17. MR 1055707, DOI 10.1007/BF01231493
- Frank G. Garvan, Some congruences for partitions that are $p$-cores, Proc. London Math. Soc. (3) 66 (1993), no. 3, 449–478. MR 1207544, DOI 10.1112/plms/s3-66.3.449
- Daniel Gorenstein, Finite simple groups, University Series in Mathematics, Plenum Publishing Corp., New York, 1982. An introduction to their classification. MR 698782, DOI 10.1007/978-1-4684-8497-7
- I. Martin Isaacs, Character theory of finite groups, Pure and Applied Mathematics, No. 69, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR 0460423
- Gordon James and Adalbert Kerber, The representation theory of the symmetric group, Encyclopedia of Mathematics and its Applications, vol. 16, Addison-Wesley Publishing Co., Reading, Mass., 1981. With a foreword by P. M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
- A. A. Klyachko, Modular forms and representations of symmetric groups, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 116 (1982), 74–85, 162 (Russian). Integral lattices and finite linear groups. MR 687842
- Burkhard Külshammer, Landau’s theorem for $p$-blocks of $p$-solvable groups, J. Reine Angew. Math. 404 (1990), 189–191. MR 1037437, DOI 10.1515/crll.1990.404.189
- Neal Koblitz, Introduction to elliptic curves and modular forms, Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1984. MR 766911, DOI 10.1007/978-1-4684-0255-1
- Gerhard O. Michler, A finite simple group of Lie type has $p$-blocks with different defects, $p\not =2$, J. Algebra 104 (1986), no. 2, 220–230. MR 866772, DOI 10.1016/0021-8693(86)90212-7
- Toshitsune Miyake, Modular forms, Springer-Verlag, Berlin, 1989. Translated from the Japanese by Yoshitaka Maeda. MR 1021004, DOI 10.1007/3-540-29593-3
- Jørn B. Olsson, On the $p$-blocks of symmetric and alternating groups and their covering groups, J. Algebra 128 (1990), no. 1, 188–213. MR 1031917, DOI 10.1016/0021-8693(90)90049-T
- Ken Ono, On the positivity of the number of $t$-core partitions, Acta Arith. 66 (1994), no. 3, 221–228. MR 1276989, DOI 10.4064/aa-66-3-221-228
- L. Kantorovitch, The method of successive approximations for functional equations, Acta Math. 71 (1939), 63–97. MR 95, DOI 10.1007/BF02547750
- G. de B. Robinson, Representation theory of the symmetric group, Mathematical Expositions, No. 12, University of Toronto Press, Toronto, 1961. MR 0125885
- Geoffrey R. Robinson, The number of blocks with a given defect group, J. Algebra 84 (1983), no. 2, 493–502. MR 723405, DOI 10.1016/0021-8693(83)90091-1
- Jean-Pierre Serre, Divisibilité des coefficients des formes modulaires de poids entier, C. R. Acad. Sci. Paris Sér. A 279 (1974), 679–682 (French). MR 382172
- Jacob Sturm, On the congruence of modular forms, Number theory (New York, 1984–1985) Lecture Notes in Math., vol. 1240, Springer, Berlin, 1987, pp. 275–280. MR 894516, DOI 10.1007/BFb0072985
- S. Minakshi Sundaram, On non-linear partial differential equations of the hyperbolic type, Proc. Indian Acad. Sci., Sect. A. 9 (1939), 495–503. MR 0000089, DOI 10.1007/BF03046994
Additional Information
- Andrew Granville
- Affiliation: address Department of Mathematics, The University of Georgia, Athens, Georgia 30602
- MR Author ID: 76180
- ORCID: 0000-0001-8088-1247
- Email: andrew@sophie.math.uga.edu
- Ken Ono
- Affiliation: address Department of Mathematics, The University of Georgia, Athens, Georgia 30602
- Address at time of publication: School of Mathematics, Institute of Advanced Study, Princeton, New Jersey 08540
- MR Author ID: 342109
- Email: ono@symcom.math.uiuc.edu
- Received by editor(s): October 18, 1994
- Received by editor(s) in revised form: February 27, 1995
- Additional Notes: The first author is a Presidential Faculty Fellow and an Alfred P. Sloan Research Fellow. His research is supported in part by the National Science Foundation
- © Copyright 1996 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 348 (1996), 331-347
- MSC (1991): Primary 20C20; Secondary 11F30, 11F33, 11D09
- DOI: https://doi.org/10.1090/S0002-9947-96-01481-X
- MathSciNet review: 1321575