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

Abstract

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 $C_n \leq S_n$. Analogous results are obtained for the alternating group $A_n$.