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

TL;DR

This paper characterizes when dihedral branching coefficients for symmetric and alternating groups are nonzero, providing bounds and generalizing positivity results for cyclic subgroups.

Contribution

It precisely determines nonzero conditions for dihedral branching coefficients and extends positivity results from cyclic to dihedral subgroups in symmetric and alternating groups.

Findings

Nonzero conditions for dihedral branching coefficients are fully characterized.

Uniform linear lower bounds established outside finite exceptions.

Positivity results are generalized from cyclic to dihedral subgroups.

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 $A_n$.