Conjectures of Donovan and Puig for Principal 3-Blocks with Abelian Defect Groups

Abstract

One of the most important and interesting conjectures in representation theory of finite groups is Donovan's conjecture. It is on block algebras of group algebras of finite groups over an algebraically closed field of prime characteristic p. Donovan conjectures that, for any given finite p-group D, up to Morita equivalence, there are only finitely many block algebras with defect group D. We prove in this article that Donovan's conjecture is true for principal block algebras in the case where D is an arbitrary finite abelian 3-group. There is another important and interesting conjecture due to Puig, which is stated as the above conjecture if we replace “Morita equivalence” by “Puig equivalence”. We prove in this paper that Puig's conjecture is true for principal 3-block algebras of finite groups with an elementary abelian Sylow 3-subgroup of order 9. Actually, we prove even more. It is proven in this paper also that there are exactly 22 non-Puig equivalent classes of principal block algebras of finite groups over a complete discrete valuation ring of rank one if a Sylow 3- subgroup D of them is an elementary abelian 3-group of order 9. The results here depend on the classification of finite simple groups.

Key Words:

• Morita equivalence
• Donovan's conjecture
• Puig equivalence
• Puig's conjecture
• Block algebra
• Group algebra
• Defect group

Acknowledgments

The author thanks Paul Fong for sending a manuscript(Fong,1996) on finite groups with abelian Sylow 3-subgroups, and Markus Linckelmann for useful and important comments. The author also thanks the referee, who read the first version of this paper carefully.

This work was supported by the JSPS (Japan Society for Promotion of Science), Grant-in-Aid for Scientific Research (Tokubetsu Kenkyuin Shorei-hi) 01016, 2001–2002; and Grant-in-Aid for Scientific Research C (2) 14540009, 2002.

Dedicated to Professor Hirouki Tachikawa on his 70th birthday.