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

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

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 .

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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