Every irreducible character of a Sylow p-subgroup of a symmetric group is encoded by a labeled tree function, and the normalizer and Galois actions become simple relabeling and permutation rules on those functions.
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On the local representation theory of symmetric groups
Every irreducible character of a Sylow p-subgroup of a symmetric group is encoded by a labeled tree function, and the normalizer and Galois actions become simple relabeling and permutation rules on those functions.