On the Positivity of Dihedral Branching Coefficients of the Symmetric and Alternating Groups

We determine precisely when the branching coefficients arising from the restriction of irreducible representations of the symmetric group $S_n$ to the dihedral subgroup $D_n$ are nonzero, and we establish uniform linear lower bounds outside a finite exceptional family. As a consequence, we recover and substantially generalize known positivity results for cyclic subgroups $C_n\le S_n$. Analogous results are obtained for the alternating group $A_n$.