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The wreath matrix

Jan Petr, Pavel Turek

TL;DR

This work introduces the wreath matrix $M(n,k)$ as a novel algebraic framework for Baranyai's wreath conjecture, recasting the problem of decomposing $\mathbb{Z}_n^{(k)}$ into wreaths in terms of spectral properties of $M$. By exploiting the natural $S_n$-action and the representation theory of symmetric groups (via Specht modules and Schur's lemma), the authors determine the eigenstructure of $M$, proving it is positive semidefinite and computing $k+1$ eigenvalues with explicit multiplicities. They also connect the matrix to a graph-theoretic setting through a related pseudoadjacency and the Delsarte--Hoffman bound, linking spectral data to independent-set considerations. The paper provides concrete kernel constructions, including $x_a$ and $y_{a,\mathcal{F}}$, and shows how Frobenius reciprocity yields a broad family of kernel vectors, offering a new algebraic route toward resolving the wreath conjecture, with detailed formulas for eigenvalues in coprime and divisible cases and several open questions about eigenvalue distinctness.

Abstract

Let $k\leq n$ be positive integers and $\mathbb{Z}_{n}$ be the set of integers modulo $n$. A conjecture of Baranyai from 1974 asks for a decomposition of $k$-element subsets of $\mathbb{Z}_{n}$ into particular families of sets called "wreaths". We approach this conjecture from a new algebraic angle by introducing the key object of this paper, the wreath matrix $M$. As our first result, we establish that Baranyai's conjecture is equivalent to the existence of a particular vector in the kernel of $M$. We then employ results from representation theory to study $M$ and its spectrum in detail. In particular, we find all eigenvalues of $M$ and their multiplicities, and identify several families of vectors which lie in the kernel of $M$.

The wreath matrix

TL;DR

This work introduces the wreath matrix as a novel algebraic framework for Baranyai's wreath conjecture, recasting the problem of decomposing into wreaths in terms of spectral properties of . By exploiting the natural -action and the representation theory of symmetric groups (via Specht modules and Schur's lemma), the authors determine the eigenstructure of , proving it is positive semidefinite and computing eigenvalues with explicit multiplicities. They also connect the matrix to a graph-theoretic setting through a related pseudoadjacency and the Delsarte--Hoffman bound, linking spectral data to independent-set considerations. The paper provides concrete kernel constructions, including and , and shows how Frobenius reciprocity yields a broad family of kernel vectors, offering a new algebraic route toward resolving the wreath conjecture, with detailed formulas for eigenvalues in coprime and divisible cases and several open questions about eigenvalue distinctness.

Abstract

Let be positive integers and be the set of integers modulo . A conjecture of Baranyai from 1974 asks for a decomposition of -element subsets of into particular families of sets called "wreaths". We approach this conjecture from a new algebraic angle by introducing the key object of this paper, the wreath matrix . As our first result, we establish that Baranyai's conjecture is equivalent to the existence of a particular vector in the kernel of . We then employ results from representation theory to study and its spectrum in detail. In particular, we find all eigenvalues of and their multiplicities, and identify several families of vectors which lie in the kernel of .
Paper Structure (6 sections, 20 theorems, 33 equations, 3 figures)

This paper contains 6 sections, 20 theorems, 33 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The wreath matrix
  3. Action of the symmetric group
  4. Computing eigenvalues
  5. Kernel of $M$
  6. Coefficients $b_{l,j}$

Key Result

Theorem 1.1

Let $n$ and $k$ be two positive integers such that $k\mid n$. Then there exists a partition of the hyperedges of a complete $k$-uniform hypergraph on $n$ vertices into perfect matchings (that is, sets of hyperedges such that each vertex lies in exactly one of these hyperedges).

Figures (3)

  • Figure 1: The $(n,k,\pi)$-wreaths for $n=10$, $\pi=(10\space5\space3\space9\space1\space6\space2\space7\space4\space8)$ and $k\in \{3,2,4\}$.
  • Figure 2: Wreath $\mathcal{F}' = \mathcal{F}_{\pi'}$ used in the proof of \ref{['le:nonzero']}. Here we take $\pi'(1) = l +1$.
  • Figure 3: The relations between the introduced $\mathbb{C} S_n$-modules.

Theorems & Definitions (43)

  • Theorem 1.1
  • Conjecture 1.2
  • Remark 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • ...and 33 more