Defect zero blocks for finite simple groups

Authors:
Andrew Granville and Ken Ono

Journal:
Trans. Amer. Math. Soc. **348** (1996), 331-347

MSC (1991):
Primary 20C20; Secondary 11F30, 11F33, 11D09

MathSciNet review:
1321575

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Abstract: We classify those finite simple groups whose Brauer graph (or decomposition matrix) has a -block with defect 0, completing an investigation of many authors. The only finite simple groups whose defect zero blocks remained unclassified were the alternating groups . Here we show that these all have a -block with defect 0 for every prime . This follows from proving the same result for every symmetric group , which in turn follows as a consequence of the *-core partition conjecture*, that every non-negative integer possesses at least one -core partition, for any . For , 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 , that was not covered in previous work, was the case . This we prove with a very different argument, by interpreting the generating function for -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 -blocks of , proving a conjecture of Garvan, that establishes certain multiplicative congruences when . By using a result of Serre concerning the divisibility of coefficients of modular forms, we show that for any given prime and positive integer , the number of blocks with defect 0 in is a multiple of for almost all . We also establish that any given prime divides the number of modularly irreducible representations of , for almost all .

**[1]**G. Almkvist,*private communication*.**[2]**George E. Andrews,*The theory of partitions*, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. Encyclopedia of Mathematics and its Applications, Vol. 2. MR**0557013****[3]**Richard Brauer,*Representations of finite groups*, Lectures on Modern Mathematics, Vol. I, Wiley, New York, 1963, pp. 133–175. MR**0178056****[4]**K. Erdmann and G. Michler,*Blocks for symmetric groups and their covering groups and quadratic forms*.**[5]**Paul Fong and Bhama Srinivasan,*The blocks of finite classical groups*, J. Reine Angew. Math.**396**(1989), 122–191. MR**988550****[6]**Frank Garvan, Dongsu Kim, and Dennis Stanton,*Cranks and 𝑡-cores*, Invent. Math.**101**(1990), no. 1, 1–17. MR**1055707**, 10.1007/BF01231493**[7]**Frank G. Garvan,*Some congruences for partitions that are 𝑝-cores*, Proc. London Math. Soc. (3)**66**(1993), no. 3, 449–478. MR**1207544**, 10.1112/plms/s3-66.3.449**[8]**Daniel Gorenstein,*Finite simple groups*, University Series in Mathematics, Plenum Publishing Corp., New York, 1982. An introduction to their classification. MR**698782****[9]**I. Martin Isaacs,*Character theory of finite groups*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Pure and Applied Mathematics, No. 69. MR**0460423****[10]**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****[11]**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****[12]**Burkhard Külshammer,*Landau’s theorem for 𝑝-blocks of 𝑝-solvable groups*, J. Reine Angew. Math.**404**(1990), 189–191. MR**1037437**, 10.1515/crll.1990.404.189**[13]**Neal Koblitz,*Introduction to elliptic curves and modular forms*, Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1984. MR**766911****[14]**Gerhard O. Michler,*A finite simple group of Lie type has 𝑝-blocks with different defects, 𝑝̸=2*, J. Algebra**104**(1986), no. 2, 220–230. MR**866772**, 10.1016/0021-8693(86)90212-7**[15]**Toshitsune Miyake,*Modular forms*, Springer-Verlag, Berlin, 1989. Translated from the Japanese by Yoshitaka Maeda. MR**1021004****[16]**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**, 10.1016/0021-8693(90)90049-T**[17]**Ken Ono,*On the positivity of the number of 𝑡-core partitions*, Acta Arith.**66**(1994), no. 3, 221–228. MR**1276989****[18]**------,*A note on the number of core partitions*, The Rocky Mtn. J. of Math (to appear).**[19]**G. de B. Robinson,*Representation theory of the symmetric group*, Mathematical Expositions, No. 12. University of Toronto Press, Toronto, 1961. MR**0125885****[20]**Geoffrey R. Robinson,*The number of blocks with a given defect group*, J. Algebra**84**(1983), no. 2, 493–502. MR**723405**, 10.1016/0021-8693(83)90091-1**[21]**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**0382172****[22]**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**, 10.1007/BFb0072985**[23]**W. Willems,*Blocks of defect zero in finite simple groups of Lie type*, J. Algebra**113**(1988), 511--522, MR**89c:2005**.

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Additional Information

**Andrew Granville**

Affiliation:
address Department of Mathematics, The University of Georgia, Athens, Georgia 30602

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

Email:
ono@symcom.math.uiuc.edu

DOI:
https://doi.org/10.1090/S0002-9947-96-01481-X

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

Article copyright:
© Copyright 1996
American Mathematical Society