The wreath matrix

TL;DR

This paper introduces the wreath matrix to analyze Baranyai's conjecture on decomposing subsets of cyclic groups, using algebraic and representation theory methods to study its spectrum and kernel.

Contribution

It establishes the equivalence of Baranyai's conjecture to a kernel vector existence in the wreath matrix and analyzes its spectrum using representation theory.

Findings

Eigenvalues and their multiplicities of the wreath matrix are fully characterized.

Several families of kernel vectors of the wreath matrix are identified.

The conjecture is reformulated as a problem about the wreath matrix's kernel.

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 $M$.