SYLOW BRANCHING TREES FOR SYMMETRIC GROUPS

. Let p >= 5 be a prime and let P be a Sylow p-subgroup of a finite symmetric group Sn. To every irreducible character of P we associate a collection of labelled, complete p-ary trees. The main results of this article describe the positivity of Sylow branching coefficients for all irreducible characters of P in terms of combinatorial properties of these trees, extending previous work on the linear characters of P.