Morita equivalence for blocks of the Schur algebras
If \(E\) is an \(n\)-dimensional \(k\)-vector space, then the tensor power \(E^{\otimes r}\) is a module for \(k \Sigma_r\), where \(\Sigma_r\) is the symmetric group of degree \(r\). The Schur algebra \(S(n,r)\) is the endomorphism ring \(\text{End}_{k \Sigma_r} (\bigoplus E^{\otimes r})\), and for a block \(B\) of \(k \Sigma_r\), denote \(S(n, r)_B = \text{End} (\bigoplus_{E_i \in B} m_i E_i)\), where \(E^{\otimes r} = \bigoplus m_i E_i\) is a decomposition into indecomposable \(k \Sigma_r\)-modules. The main result of this paper states that for a fixed weight \(w\), the number of Morita equivalence classes of algebras of the form \(S(n,r)_B\), where \(B\) is a block of \(k \Sigma_r\) of weight \(w\), is finite. This theorem is an analogue for Schur algebras of the main result of \textit{J. Scopes} [J. Algebra 142, 441-455 (1991; Zbl 0736.20008)], where Donovan's conjecture was verified for blocks of symmetric groups. Scopes' techniques and a detailed analysis of Young modules are used in the proof of the theorem.
scientific article; zbMATH DE number 645082