Abstract. Donovan’s conjecture suggests that for a given integer n, there are finitely many isomorphism classes of basic algebras of dimension n which are Morita equivalent to a block algebra of a finite group algebra. That is, we would expect to see very few algebras that arise as blocks of finite group algebras. In [1] Linckelmann confirms this is the case for n ≤ 12, by classifying basic algebras of blocks of finite group algebras (over an algebraically closed field) of dimension at most 12, with one exception. At each dimension this classification begins by computing all possible Cartan matrices of hypothetical algebras that can arise as basic algebras of blocks of finite groups. At dimension 9, one particular Cartan matrix suggested an algebra which we denote A, for which it was not then known whether it could arise as such a block. In our work, we rule out that this final case, showing that A cannot arise as a basic algebra of a block of a finite group algebra. In order to do so, we give an explicit description of A, in particular computing its stable centre Z(A). In the literature there are two possible candidates for a block of a finite group with the same Cartan matrix as A, and since stable centres are invariant under stable equivalences of Morita type, we use Z(A) to show that A is not the basic algebra of either of these blocks.

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