Strong Gelfand pairs of symmetric groups. (English) Zbl 1535.20072
Summary: A strong Gelfand pair is a pair \((G,H)\), \(H\leq G\), of finite groups such that the Schur ring determined by the \(H\)-classes \(g^H\), \(g\in G\), is a commutative ring. We find all strong Gelfand pairs \((S_n,H)\). We also define an extra strong Gelfand pair \((G,H)\), this being a strong Gelfand pair of maximal dimension, and show that in this case \(H\) must be abelian.
MSC:
| 20C30 | Representations of finite symmetric groups |
| 20C15 | Ordinary representations and characters |
| 20E22 | Extensions, wreath products, and other compositions of groups |
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