Dade's conjecture for tame blocks
Katsuhiro Uno · Osaka City University (Osaka City University) · 1994
UNOin Section 1, we review several results on tame blocks in Section 2. Most of them are found in [4] or [8], or just easy exercises.Using the results in Section 2 and some local analysis, we determine, in Section 3, the related blocks which contribute the alternating sum.Section 4 is devoted to studying some actions of automorphisms on Irr(2?) and on the column index set of the generalized decomposition matrix of B. Through Brauer's permutation lemma, one can see the relation between these two actions.This may give a device for counting the number of invariant elements in Irr(i?) in terms of the action on local objects such as subsections.However, we must also consider the action on IBr(i?) since irreducible Brauer characters appear as the column indices as well.This is analyzed by, with Erdmann's classification of tame algebras, looking at the actions on the ordinary decomposition matrices, on the Cartan matrices or on the stable Auslander-Reiten quivers of the modular block algebras.Eventually, the number of invariant ordinary irreducible characters is given completely in terms of the actions on local objects.These are done in Section 5. We complete the proof of our main result in Section 6.ACKNOWLEDGEMENT.The first attempt to the subject in this paper was to prove that the ordinary conjecture (the simplest form) holds for tame blocks.It was Professor Dade who suggested the author that the extended conjecture for tame blocks should also be handled.Thus, the author would like to express his heartfelt gratitude to