A remark on Donovan's conjecture. (Q1881571)

Let \(p\) be a prime number, \(k\) an algebraically closed field, and \(P\) a fixed finite \(p\)-group. A conjecture of Donovan's states that up to Morita equivalence there are only finitely many \(k\)-algebras that occur as blocks with defect group \(P\) of group algebras \(kG\), where \(G\) is a finite group. This would imply Brauer's Problem 22: there are only finitely many possibilities for the Cartan entries of such blocks. Now consider the Frobenius automorphism of \(k\), which induces a permutation of the blocks of \(kG\), and denote by \(m(G,B)\) the number of Morita equivalence classes of \(k\)-algebras in the set \(\{\sigma^d(B)\mid d\in\mathbb{N}\}\). The author conjectures that there is an integer \(m\), depending only on \(P\), such that \(m(G,B)\leq m\) for all finite groups \(G\) and all blocks \(B\) with defect group \(P\) of \(kG\). Then she proves that the last two conjectures are together equivalent to Donovan's conjecture.