On the Positivity of Dihedral Branching Coefficients of the Symmetric and Alternating Groups (2512.14381v1)

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 $A_n$.

• The paper establishes explicit conditions under which dihedral branching coefficients are positive, providing linear lower bounds for Sₙ and Aₙ restrictions.
• It employs a combination of inductive combinatorial techniques, Littlewood–Richardson theory, and computational verification to analyze partition families and exceptional cases.
• The results refine decomposition counts in relevant Hecke algebras and suggest pathways for extending similar analyses to other nonabelian subgroup restrictions.

Positivity of Dihedral Branching Coefficients in Symmetric and Alternating Groups

Introduction and Context

The decomposition of irreducible representations of $S_n$ upon restriction to subgroups is a central problem in algebraic combinatorics and finite group representation theory. While much is known for subgroups such as Young, Sylow-$p$, and cyclic subgroups—especially those generated by $n$-cycles—systematic understanding for non-cyclic, non-Young subgroups is underdeveloped. Dihedral subgroups $D_n$, the smallest nonabelian subgroups containing an $n$-cycle, are particularly significant due to their minimal complexity beyond cyclics. This work gives a complete characterization of the positivity of the multiplicities ("branching coefficients") of irreducible representations in the restriction of $S_n$ (or $A_n$) representations to $D_n$, together with explicit linear lower bounds outside of explicit, finite exceptional families.

Main Results

Dihedral Branching Coefficients

The core outcome is an explicit, case-by-case determination of when the multiplicities $\langle \Res^{S_n}_{D_n} \chi_\lambda, \psi \rangle$ are zero or positive as $\lambda$ varies over partitions of $n$. The results are organized according to the parity and arithmetic properties of $n$, and further to the structure of the involved irreducible characters of $D_n$ (linear and degree-$2$ characters).

A key finding is that, for $n\geq 11$, apart from an explicit small set of exceptional partitions (which are classified in detail for each parity pattern), all multiplicities are uniformly lower-bounded by linear functions in $n$: either $n/6$ or $n/12$, depending on the situation. For instance, for most partitions, it is shown that

• $d_{\psi}^j(\lambda)>\tfrac{n}{6}$ for all $j$
• $$d_{\mathbb{1}}^{\pm\mathbb{1}}(\lambda) > \tfrac{n}{12}$$

where $d_{\psi}^j$ is the multiplicity of the $j$th irreducible degree-$2$ character and $d_{\mathbb{1}}^{\pm\mathbb{1}}$ for the relevant linear character.

The complete statement organizes these exceptions by partition shape, manifesting that only hooks, the all-one-part and one-part partitions, certain two-row and double hook shapes, and the rectangular partition $(n/2,n/2)$ (when $n$ is even) can have multiplicity zero or one.

Connection to Cyclic Branching and Major Index

The analysis subsumes and strengthens the known results for cyclic subgroups. The branching coefficients for restriction to $C_n$ (the cyclic group generated by an $n$-cycle) admit combinatorial interpretation in terms of the "major index" statistic on standard Young tableaux ("SYT"), as established in previous works (e.g., [Klyachko], [Kraskiewicz-Weyman]):

$a_\lambda^r = |\{T\in \SYT(\lambda) \mid \maj(T) \equiv r \bmod n\}|$

The present work extends the general lower bounds known for these coefficients and explains their extensions to the dihedral case via explicit relationships among the branching coefficients, the eigenvalue support structures, and the symmetry of these statistics.

Alternating Groups

Analogous complete results are given for the restriction of irreducible $A_n$ modules to dihedral subgroups for odd $n$, with all exceptions and linear lower bounds classified as in the symmetric group case.

Technical Approach

The main proofs combine inductive combinatorial constructions, Littlewood–Richardson theory, and careful analysis of the symmetric function expansions of relevant branching rules.

A significant combinatorial effort is made to cover all partition families not affected by direct computation for small $n$. For large $n$, multi-layered induction arguments on the structure of partitions are deployed, notably through:

• Iterative application of partition dominance and LR rules
• Embedding arguments for wreath product subgroups and their action on Young diagrams
• Explicit use of Murnaghan-Nakayama and centralizer techniques for character bounds

SageMath-based computational verification is used for small $n$, ensuring that all exceptional cases are covered.

Consequences and Implications

The main theoretical implication is a sharp understanding of which irreducible components ever appear in the restriction to $D_n$ and with what minimal multiplicity, demarcating the boundary between "ubiquitous" and "sporadic" behavior of dihedral coefficient positivity.

Practically, these results immediately provide:

• Exact counts and explicit lower bounds for irreducible components in the associated Hecke algebras $\mathcal{H}(S_n, D_n, \chi)$.
• Improved linear bounds compared to classical estimates (e.g., Kovács–Stöhr bounds for free Lie modules), which may impact asymptotics for related functionals and characters.
• An explicit list of exceptional "missing" irreducibles which supports algorithmic enumeration efforts in computational representation theory.

Open Problems

Despite the breadth of these results, the paper raises further combinatorial questions, in particular:

• Is there a statistic on SYT refining the major index, specializing to the dihedral setting, that exactly describes the dihedral branching coefficients for the linear characters of $D_n$?
• How can one fully describe the combinatorics underlying dimensions of generalized Hecke algebras corresponding to subgroups like $D_n$?

Addressing these questions would deepen the combinatorial understanding of non-abelian symmetry restrictions and their enumerative invariants.

Future Directions

Future theoretical developments may:

• Extend analogous positivity and lower-bound results to other non-abelian, non-Young subgroups of $S_n$ and $A_n$, possibly via similar combinatorial and inductive approaches.
• Seek optimal universal or partition-dependent bounds, potentially improving the linear coefficients.
• Provide detailed combinatorial bijections for dihedral coefficients, clearly refining the major index or other symmetric group parameters.
• Explore direct connections with modular representation theory, especially in relation to Hecke algebra representations and categorification frameworks.

Algorithmically, explicit knowledge of non-vanishing in these restriction problems directly informs computation of decomposition matrices, character tables, and branching graphs for large $n$.

Conclusion

This work offers a detailed, explicit, and nearly exhaustive answer to the question of when dihedral branching coefficients from $S_n$ (and $A_n$) are nonzero, complemented by strong uniform lower bounds. By unifying classical combinatorial invariant (major index) perspectives with detailed branching analysis, it significantly extends the understanding of representation restriction phenomena with respect to minimal nonabelian subgroups. The explicit classification of exceptional partitions and lower bounds opens further questions regarding analogous combinatorial statistics and refinement, suggesting rich future interactions between algebraic combinatorics and representation theory.