
Using the techniques of the author's paper on Morita equivalence of blocks of symmetric groups the following strong results are proved for blocks of defect 2 in symmetric groups (corresponding results hold for the alternating groups). Theorem I. A block of a symmetric group with defect two in characteristic \(p > 0\) has the following properties. 1. All the decomposition numbers are 0 or 1. 2. A row in the decomposition matrix has at most 5 non-zero entries. 3. All diagonal entries in the Cartan matrix are greater than or equal to 3. 4. All non-diagonal entries in the Cartan matrix are less than or equal to 2. 5. The dimension of \(\text{Ext}^1 [D^\lambda, D^\mu]\) is 0 or 1 for all \(p\)-regular \(\lambda\) and \(\mu\). 6. The dimension of \(\text{Ext}^1 [D^\lambda, D^\mu]\) is 0 if \(D^\mu\) is isomorphic to \(D^\lambda\) or \(D^\lambda \otimes \text{alt}\). 7. All the projective indecomposable modules have Loewy length 5 and are stable, that is their Loewy series are the same as their socle series. Theorem II. Given any symmetric group block \(B\) of defect two there is an algorithm which one may apply to determine 1. the decomposition matrix of \(B\); 2. the Cartan matrix of \(B\); 3. the \(\text{Ext}^1\) quiver of \(B\); 4. the Loewy structures of the projective indecomposable modules of \(B\).
Loewy series, Modular representations and characters, Representations of finite symmetric groups, Loewy length, blocks of defect 2, projective indecomposable modules, decomposition numbers, Cartan matrix, blocks of symmetric groups, alternating groups, decomposition matrix
Loewy series, Modular representations and characters, Representations of finite symmetric groups, Loewy length, blocks of defect 2, projective indecomposable modules, decomposition numbers, Cartan matrix, blocks of symmetric groups, alternating groups, decomposition matrix
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