Defect zero blocks for finite simple groups
Authors:
Andrew Granville and Ken Ono
Journal:
Trans. Amer. Math. Soc. 348 (1996), 331347
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 nonnegative integer possesses at least one core partition, for any . For , we reduce this problem to Lagrange's Theorem that every nonnegative 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, AddisonWesley
Publishing Co., Reading, Mass.LondonAmsterdam, 1976. Encyclopedia of
Mathematics and its Applications, Vol. 2. MR 0557013
(58 #27738)
 [3]
Richard
Brauer, Representations of finite groups, Lectures on Modern
Mathematics, Vol. I, Wiley, New York, 1963, pp. 133–175. MR 0178056
(31 #2314)
 [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
(90f:20065)
 [6]
Frank
Garvan, Dongsu
Kim, and Dennis
Stanton, Cranks and 𝑡cores, Invent. Math.
101 (1990), no. 1, 1–17. MR 1055707
(91h:11106), http://dx.doi.org/10.1007/BF01231493
 [7]
Frank
G. Garvan, Some congruences for partitions that are
𝑝cores, Proc. London Math. Soc. (3) 66
(1993), no. 3, 449–478. MR 1207544
(94c:11101), http://dx.doi.org/10.1112/plms/s366.3.449
 [8]
Daniel
Gorenstein, Finite simple groups, University Series in
Mathematics, Plenum Publishing Corp., New York, 1982. An introduction to
their classification. MR 698782
(84j:20002)
 [9]
I.
Martin Isaacs, Character theory of finite groups, Academic
Press [Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1976. Pure
and Applied Mathematics, No. 69. MR 0460423
(57 #417)
 [10]
Gordon
James and Adalbert
Kerber, The representation theory of the symmetric group,
Encyclopedia of Mathematics and its Applications, vol. 16,
AddisonWesley Publishing Co., Reading, Mass., 1981. With a foreword by P.
M. Cohn; With an introduction by Gilbert de B. Robinson. MR 644144
(83k:20003)
 [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
(85f:11034)
 [12]
Burkhard
Külshammer, Landau’s theorem for 𝑝blocks of
𝑝solvable groups, J. Reine Angew. Math. 404
(1990), 189–191. MR 1037437
(91c:20018), http://dx.doi.org/10.1515/crll.1990.404.189
 [13]
Neal
Koblitz, Introduction to elliptic curves and modular forms,
Graduate Texts in Mathematics, vol. 97, SpringerVerlag, New York,
1984. MR
766911 (86c:11040)
 [14]
Gerhard
O. Michler, A finite simple group of Lie type has 𝑝blocks
with different defects, 𝑝̸=2, J. Algebra
104 (1986), no. 2, 220–230. MR 866772
(87m:20038), http://dx.doi.org/10.1016/00218693(86)902127
 [15]
Toshitsune
Miyake, Modular forms, SpringerVerlag, Berlin, 1989.
Translated from the Japanese by Yoshitaka Maeda. MR 1021004
(90m:11062)
 [16]
Jørn
B. Olsson, On the 𝑝blocks of symmetric and alternating
groups and their covering groups, J. Algebra 128
(1990), no. 1, 188–213. MR 1031917
(90k:20022), http://dx.doi.org/10.1016/00218693(90)90049T
 [17]
Ken
Ono, On the positivity of the number of 𝑡core
partitions, Acta Arith. 66 (1994), no. 3,
221–228. MR 1276989
(95a:11092)
 [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 (23 #A3182)
 [20]
Geoffrey
R. Robinson, The number of blocks with a given defect group,
J. Algebra 84 (1983), no. 2, 493–502. MR 723405
(85c:20009), http://dx.doi.org/10.1016/00218693(83)900911
 [21]
JeanPierre
Serre, Divisibilité des coefficients des formes modulaires
de poids entier, C. R. Acad. Sci. Paris Sér. A
279 (1974), 679–682 (French). MR 0382172
(52 #3060)
 [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
(88h:11031), http://dx.doi.org/10.1007/BFb0072985
 [23]
W. Willems, Blocks of defect zero in finite simple groups of Lie type, J. Algebra 113 (1988), 511522, MR 89c:2005.
 [1]
 G. Almkvist, private communication.
 [2]
 G. Andrews, The theory of partitions, Encyclopedia of Mathematics and its Applications, AddisonWesley, Reading, 1976, MR 58:27738.
 [3]
 R. Brauer, Representations of finite groups, Lect. on Modern Math. 1 (1963), 133175, MR 31:2314.
 [4]
 K. Erdmann and G. Michler, Blocks for symmetric groups and their covering groups and quadratic forms.
 [5]
 P. Fong and B. Srinivasan, The blocks of finite classical groups, J. reine angew. Math. 396 (1989), 121191, MR 90f:20065.
 [6]
 F. Garvan, D. Kim and D. Stanton, Cranks and cores, Invent. Math. 101 (1990), 117, MR 91h:11106.
 [7]
 F. Garvan, Some congruence properties for partitions that are cores, Proc. London Math. Soc. (3) 66 (1993), 449478, MR 94c:11101.
 [8]
 D. Gorenstein, Finite simple groups: An introduction to their classification, Plenum Press, New York and London, 1982, MR 84j:20002.
 [9]
 I. M. Isaacs, Character theory of finite groups, Academic Press, New York, 1976, MR 57:417.
 [10]
 G. James and A. Kerber, The representation theory of the symmetric group, AddisonWesley, Reading, 1981, MR 83k:20003.
 [11]
 A. Klyachko, Modular forms and representations of symmetric groups, integral lattices and finite linear groups, Zap. Nauchn. Sem. Leningrad Otdel. Mat. Inst. Steklov 116 (1982), MR 85f:11034.
 [12]
 B. Külshammer, Landau's theorem for blocks of solvable groups, J. reine angew. Math. 404 (1990), 171188, MR 91c:20018.
 [13]
 N. Koblitz, Introduction to elliptic curves and modular forms, SpringerVerlag, New York, 1984, MR 86c:11040.
 [14]
 G. Michler, A finite simple group of Lie type has blocks with different defects if , J. Algebra 104 (1986), 220230, MR 87m:20038.
 [15]
 T. Miyake, Modular forms, SpringerVerlag, New York, 1989, MR 90m:11062.
 [16]
 J. Olsson, On the blocks of symmetric and alternating groups and their covering groups, J. Algebra 128 (1990), 188213, MR 90k:20022.
 [17]
 K. Ono, On the positivity of the number of core partitions, Acta Arithmetica 66 (1994), 221228, MR 95a:11092.
 [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, Edinburgh Univ. Press, 1961, MR 23:A3182.
 [20]
 G. Robinson, The number of blocks with a given defect group, J. Algebra 84 (1983), 493502, MR 85c:20009.
 [21]
 J.P. Serre, Divisibilite des coefficients des formes modulaires de poids entier, C.R. Acad. Sci. Paris A 279 (1974), 679682, MR 52:3060.
 [22]
 J. Sturm, On the congruence of modular forms, Springer Lect. Notes in Math. 1240, Springer Verlag, New York, 1984, pp. (275280), MR 88h:11031.
 [23]
 W. Willems, Blocks of defect zero in finite simple groups of Lie type, J. Algebra 113 (1988), 511522, 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:
http://dx.doi.org/10.1090/S000299479601481X
PII:
S 00029947(96)01481X
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
