The essential rank of fusion systems of blocks of symmetric groups. (English) Zbl 1260.20014
Summary: Let \(p\) be a prime and let \(B\) be a \(p\)-block of the symmetric group \(S(n)\) on \(n\) points. Let \((D,b_D)\) be a Sylow \(B\)-subgroup of \(S(n)\). We consider the fusion system \(\mathcal F_{(D,b_D)}(G,B)\) and determine a precise formula for its essential rank. In addition, the \(p\)-blocks \(B\) which admit a \(p\)-local subgroup of \(S(n)\) controlling \(B\)-fusion are characterized.
MSC:
| 20C30 | Representations of finite symmetric groups |
| 20C20 | Modular representations and characters |
| 20B35 | Subgroups of symmetric groups |
| 20D20 | Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure |
| 55R35 | Classifying spaces of groups and \(H\)-spaces in algebraic topology |
Keywords:
symmetric groups; conjugation families; essential rank; fusion systems; blocks; \(p\)-local subgroupsReferences:
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