Intersecting Families of Permutations (1011.3342v2)

Abstract: A set of permutations $I \subset S_n$ is said to be {\em k-intersecting} if any two permutations in $I$ agree on at least $k$ points. We show that for any $k \in \mathbb{N}$, if $n$ is sufficiently large depending on $k$, then the largest $k$-intersecting subsets of $S_n$ are cosets of stabilizers of $k$ points, proving a conjecture of Deza and Frankl. We also prove a similar result concerning $k$-cross-intersecting subsets. Our proofs are based on eigenvalue techniques and the representation theory of the symmetric group.

• The paper proves that for sufficiently large n, the largest k-intersecting subsets of Sₙ are cosets of k-stabilizers, validating the Deza-Frankl conjecture.
• It establishes that for k-cross-intersecting families, the maximum product of sizes is achieved when the sets coincide, thereby resolving Leader's conjecture.
• By applying Fourier analysis on the symmetric group, the study demonstrates the stability of near-maximal intersecting sets and generalizes Birkhoff's theorem for Boolean functions.

Analysis of "Intersecting Families of Permutations"

The paper "Intersecting Families of Permutations," authored by Ellis, Friedgut, and Pilpel, addresses combinatorial questions concerning permutation groups, specifically within the symmetric group $S_n$. The paper builds upon classical theorems such as the Erdős-Ko-Rado theorem and proposes an analysis for intersecting families of permutations, extending the understanding of $k$-intersecting subsets and cosets of stabilizers in these groups.

Summary of Key Contributions

1. Main Theorem and Conjecture of Deza and Frankl: The paper primarily revolves around proving a conjecture by Deza and Frankl, which states that for sufficiently large $n$ relative to $k$, the largest $k$-intersecting subsets of $S_n$ correspond to cosets of $k$-stabilizers. Through elegant arguments involving eigenvalue techniques and representation theory, the authors establish this conjecture, describing the maximal structure of these intersecting families.
2. Cross-intersecting Families and Leader's Conjecture: Additionally, the authors address $k$-cross-intersecting subsets of $S_n$, proving that if $I, J$ are two such intersecting sets, then the product of their sizes is maximized precisely when they coincide and form a $k$-coset. This resolves a conjecture posed by Leader in the cross-intersection context.
3. Fourier Analysis on the Symmetric Group: The use of Fourier analysis, an advanced tool that penetrates various areas of combinatorics and computer science, is customized here for a non-Abelian group setting. This application is a pivotal feature of the research, allowing the decomposition of permutation representations and facilitating the proof of the induced stability result.
4. Stable Structures and Unique Maximizers: The paper ensures not only the characterization of maximal structures but also discusses stability. This ensures that near-maximal $k$-intersecting families approach the structure of $k$-cosets, implying a precise structural rigidity in large symmetric groups.
5. Generalization of Birkhoff's Theorem: Alongside proving their main results, the authors extend Birkhoff's theorem on bistochastic matrices to a generalized setting concerning Boolean functions on permutations, revealing a deeper combinatorial structure and further connecting combinatorial optimization with algebraic properties of permutation groups.

Implications and Future Prospects

The theoretical advances made in this paper have significant implications in both combinatorics and group theory. Practically, these results improve our understanding of symmetric group actions and provide methodologies adaptable to similar algebraic settings. The paper predicts further exploration into where non-trivial sharply $k$-transitive permutation subgroups can exist and encourages examining smaller values of $n$ or finding tighter bounds for intersecting families, especially for cases not covered by the current theory.

Furthermore, the intersection of Fourier analysis and group representation theory opens avenues for novel algorithms and techniques in computational combinatorics, potentially impacting diverse fields such as computational biology, coding theory, and network theory, where permutation problems often arise.

Conclusion

"Intersecting Families of Permutations" presents a rigorous and comprehensive exploration into the structure of permutation groups, corroborating conjectures with substantial theoretical development. By leveraging advanced mathematical concepts and pioneering a blend of algebraic and combinatorial methods, this paper forms a base for future research into intersecting families across large symmetric groups, continuing the legacy of combinatorial optimization and theory development.