## Abstract

Let k be an algebraically closed ﬁeld of characteristic p > 0, let D be a ﬁnite p-group. Brou´e’s Abelian Defect Conjecture [Bro] predicts that if D is abelian and G is a ﬁnite group with D ≤ G, and B is a block algebra of kG with defect group D, then B is derived equivalent to its Brauer correspondent, a block algebra of NG(D) which also has defect group D.

Donovan’s Conjecture predicts that there are, up to Morita equivalence, only ﬁnitely many diﬀerent block algebras of ﬁnite groups with defect group D.

Both of these conjectures are still open, although progress has been made on special cases of both. It is natural to ask (and a number of people have asked) if they are related in the following way: if it were true that there are only ﬁnitely many Morita equivalence classes of algebras derived equivalent to each block of defect D, then Brou´e’s Conjecture would imply Donovan’s Conjecture for abelian D. In some small cases, this is true. If D is cyclic, every block with defect D is Morita equivalent to a Brauer tree algebra, and it is known that the only algebras derived equivalent to a Brauer tree algebra are also Morita equivalent to Brauer tree algebras for trees with the same number of edges and same multiplicty of the exceptional vertex, so there are only ﬁnitely many possibilities.

If D is a Klein 4-group, then it is also known that there are only a very small number of Morita classes of algebras derived equivalent to any block with defect group D (and in fact all of these occur as blocks of group algebras.

However, for larger D it seems that this is very rarely true. In fact, we believe that the cases mentioned above are probably the only cases where this is true.

In this paper, we give a method of showing by a fairly simple calculation that a given block algebra has inﬁnitely many Morita equivalence classes of algebras derived equivalent to it, and show that the method applies to several blocks with small abelian defect group. In fact, our method produces a sequence of algebras with unbounded Cartan invariants (which is how we can detect that there are inﬁnitely many Morita equivalence classes. A weaker version of Donovan’s Conjecture states that the blocks with a given defect group D have bounded Cartan invariants, and this means that even this weaker version wouldn’t follow

in a straightforward way from Brou´e’s Conjecture.

Donovan’s Conjecture predicts that there are, up to Morita equivalence, only ﬁnitely many diﬀerent block algebras of ﬁnite groups with defect group D.

Both of these conjectures are still open, although progress has been made on special cases of both. It is natural to ask (and a number of people have asked) if they are related in the following way: if it were true that there are only ﬁnitely many Morita equivalence classes of algebras derived equivalent to each block of defect D, then Brou´e’s Conjecture would imply Donovan’s Conjecture for abelian D. In some small cases, this is true. If D is cyclic, every block with defect D is Morita equivalent to a Brauer tree algebra, and it is known that the only algebras derived equivalent to a Brauer tree algebra are also Morita equivalent to Brauer tree algebras for trees with the same number of edges and same multiplicty of the exceptional vertex, so there are only ﬁnitely many possibilities.

If D is a Klein 4-group, then it is also known that there are only a very small number of Morita classes of algebras derived equivalent to any block with defect group D (and in fact all of these occur as blocks of group algebras.

However, for larger D it seems that this is very rarely true. In fact, we believe that the cases mentioned above are probably the only cases where this is true.

In this paper, we give a method of showing by a fairly simple calculation that a given block algebra has inﬁnitely many Morita equivalence classes of algebras derived equivalent to it, and show that the method applies to several blocks with small abelian defect group. In fact, our method produces a sequence of algebras with unbounded Cartan invariants (which is how we can detect that there are inﬁnitely many Morita equivalence classes. A weaker version of Donovan’s Conjecture states that the blocks with a given defect group D have bounded Cartan invariants, and this means that even this weaker version wouldn’t follow

in a straightforward way from Brou´e’s Conjecture.

Original language | English |
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Type | arXiv |

Media of output | PDF, text |

Number of pages | 8 |

Publication status | Published - 9 Oct 2013 |