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.
Giannelli, McKay bijections for symmetric and alternating groups, Algebra Number Theory 15(7) (2021), 1809--1835
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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.