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Sean Eberhard

Publications and source records attributed to Sean Eberhard.

At least 37 records · Page 2Linked to original sources
arXiv2022

Hamming sandwiches

We describe primitive association schemes $\mathfrak{X}$ of degree $n$ such that $\mathrm{Aut}(\mathfrak{X})$ is imprimitive and $|\mathrm{Aut}(\mathfrak{X})| \geq \exp(n^{1/8})$, contradicting a conjecture of Babai. This and other examples we give are the first known examples of nonschurian primitive coherent configurations (PCC) with more than a quasipolynomial number of automorphisms. Our constructions are "Hamming sandwiches", association schemes sandwiched between the $d$th tensor power of the trivial scheme and the $d$-dimensional Hamming scheme. We study Hamming sandwiches in general, and exhaustively for $d \leq 8$. We revise Babai's conjecture by suggesting that any PCC with more than a quasipolynomial number of automorphisms must be an association scheme sandwiched between a tensor power of a Johnson scheme and the corresponding full Cameron scheme. If true, it follows that any nonschurian PCC has at most $\exp O(n^{1/8} \log n)$ automorphisms.

math.CO↗
arXiv2022

Probability of generation by random permutations of given cycle type

Suppose $π$ and $π'$ are two random elements of $S_n$ with constrained cycle types such that $π$ has $x n^{1/2}$ fixed points and $yn/2$ two-cycles, and likewise $π'$ has $x' n^{1/2}$ fixed points and $y'n/2$ two-cycles. We show that the events that $G = \langle π, π' \rangle$ is transitive and $G \geq A_n$ both have probability approximately \[(1 - yy')^{1/2} \exp\left(- \frac{xx' + \frac12 x^2 y' + \frac12 {x'}^2 y}{1 - yy'}\right),\] provided $(x, x')$ is not close to $(0, \infty)$ or $(\infty, 0)$. This formula is derived from some preliminary results in a recent paper (arXiv:1904.12180) of the authors. As an application, we show that two uniformly random elements of uniformly random conjugacy classes of $S_n$ generate the group with probability about 51%.

math.GR↗
arXiv2022

An asymptotic for the Hall--Paige conjecture

Hall and Paige conjectured in 1955 that a finite group $G$ has a complete mapping if and only if its Sylow $2$-subgroups are trivial or noncyclic. This conjecture was proved in 2009 by Wilcox, Evans, and Bray using the classification of finite simple groups and extensive computer algebra. Using a completely different approach motivated by the circle method from analytic number theory, we prove that the number of complete mappings of any group $G$ of order $n$ satisfying the Hall--Paige condition is $(e^{-1/2} + o(1)) \, |G^\text{ab}| \, n!^2/n^n$.

math.CO↗
arXiv2021

The characteristic polynomial of a random matrix

Form an $n \times n$ matrix by drawing entries independently from $\{\pm1\}$ (or another fixed nontrivial finitely supported distribution in $\mathbf{Z}$) and let $ϕ$ be the characteristic polynomial. Conditionally on the extended Riemann hypothesis, with high probability $ϕ$ is irreducible and $\mathrm{Gal}(ϕ) \geq A_n$.

math.NT↗
arXiv2021

Probabilistically nilpotent groups of class two

For $G$ a finite group, let $d_2(G)$ denote the proportion of triples $(x, y, z) \in G^3$ such that $[x, y, z] = 1$. We determine the structure of finite groups $G$ such that $d_2(G)$ is bounded away from zero: if $d_2(G) \geq ε> 0$, $G$ has a class-4 nilpotent normal subgroup $H$ such that $[G : H] $ and $|γ_4(H)|$ are both bounded in terms of $ε$. We also show that if $G$ is an infinite group whose commutators have boundedly many conjugates, or indeed if $G$ satisfies a certain more general commutator covering condition, then $G$ is finite-by-class-3-nilpotent-by-finite.

math.GR↗
arXiv2021

Biased multilinear maps of abelian groups

We adapt the theory of partition rank and analytic rank to the category of abelian groups. If $A_1, \dots, A_k$ are finite abelian groups and $ϕ: A_1 \times \cdots \times A_k \to \mathbf{T}$ is a multilinear map, where $\mathbf{T} = \mathbf{R}/\mathbf{Z}$, the bias of $ϕ$ is defined to be the average value of $\exp(i 2 πϕ)$. If the bias of $ϕ$ is bounded away from zero we show that $ϕ$ is the sum of boundedly many multilinear maps each of which factors through the standard multiplication map of $\mathbf{Z}/q\mathbf{Z}$ for some bounded prime power $q$. Relatedly, if $F : A_1 \times \cdots \times A_{k-1} \to B$ is a multilinear map such that $\mathbf{P}(F = 0)$ is bounded away from zero, we show that $F$ is the sum of boundedly many multilinear functions of a particular form. These structure theorems generalize work of several authors in the elementary abelian case to the arbitrary abelian case. The set of all possible biases is also investigated.

math.CO↗
arXiv2021

Babai's conjecture for high-rank classical groups with random generators

Let $G = \mathrm{SCl}_n(q)$ be a quasisimple classical group with $n$ large, and let $x_1, \dots, x_k \in G$ random, where $k \geq q^C$. We show that the diameter of the resulting Cayley graph is bounded by $q^2 n^{O(1)}$ with probability $1 - o(1)$. In the particular case $G = \mathrm{SL}_n(p)$ with $p$ a prime of bounded size, we show that the same holds for $k = 3$.

math.GR↗
arXiv2021

Permutations With Equal Orders

Let $P(n)$ be the probability that two independent, uniformly random permutations of $[n]$ have the same order, and let $K(n)$ be the probability that they are in the same conjugacy class. Answering a question of Thibault Godin, we prove that $ P(n)=n^{-2+o(1)}$ and that $\lim\sup \frac{ P(n) }{ K(n) }=\infty.$

math.CO↗
arXiv2020

Mixing time of the Chung--Diaconis--Graham random process

Define $(X_n)$ on $\mathbf{Z}/q\mathbf{Z}$ by $X_{n+1} = 2X_n + b_n$, where the steps $b_n$ are chosen independently at random from $-1, 0, +1$. The mixing time of this random walk is known to be at most $1.02 \log_2 q$ for almost all odd $q$ (Chung--Diaconis--Graham, 1987), and at least $1.004 \log_2 q$ (Hildebrand, 2008). We identify a constant $c = 1.01136\dots$ such that the mixing time is $(c+o(1))\log_2 q$ for almost all odd $q$. In general, the mixing time of the Markov chain $X_{n+1} = a X_n + b_n$ modulo $q$, where $a$ is a fixed positive integer and the steps $b_n$ are i.i.d. with some given distribution in $\mathbf{Z}$, is related to the entropy of a corresponding self-similar Cantor-like measure (such as a Bernoulli convolution). We estimate the mixing time up to a $1+o(1)$ factor whenever the entropy exceeds $(\log a)/2$.

math.PR↗
arXiv2020

Association schemes for diagonal groups

For any finite group $G$, and any positive integer $n$, we construct an association scheme which admits the diagonal group $D_n(G)$ as a group of automorphisms. The rank of the association scheme is the number of partitions of $n$ into at most $|G|$ parts, so is $p(n)$ if $|G|\ge n$; its parameters depend only on $n$ and $|G|$. For $n=2$, the association scheme is trivial, while for $n=3$ its relations are the Latin square graph associated with the Cayley table of $G$ and its complement. A transitive permutation group $G$ is said to be \emph{AS-free} if there is no non-trivial association scheme admitting $G$ as a group of automorphisms. A consequence of our construction is that an AS-free group must be either $2$-homogeneous or almost simple. We construct another association scheme, finer than the above scheme if $n>3$, from the Latin hypercube consisting of $n$-tuples of elements of $G$ with product the identity.

math.GR↗
arXiv2020

On symmetric intersecting families of vectors

A family of vectors $A \subset [k]^n$ is said to be intersecting if any two elements of $A$ agree on at least one coordinate. We prove, for fixed $k \ge 3$, that the size of a symmetric intersecting subfamily of $[k]^n$ is $o(k^n)$, which is in stark contrast to the case of the Boolean hypercube (where $k =2$). Our main contribution addresses limitations of existing technology: while there is now some spectral machinery, developed by Ellis and the third author, to tackle extremal problems in set theory involving symmetry, this machinery relies crucially on the interplay between up-sets and biased product measures on the Boolean hypercube, features that are notably absent in the problem at hand; here, we describe a method for circumventing these barriers.

math.CO↗
arXiv2019

Random generation with cycle type restrictions

We study random generation in the symmetric group when cycle type restrictions are imposed. Given $π, π' \in S_n$, we prove that $π$ and a random conjugate of $π'$ are likely to generate at least $A_n$ provided only that $π$ and $π'$ have not too many fixed points and not too many $2$-cycles. As an application, we investigate the following question: For which positive integers $m$ should we expect two random elements of order $m$ to generate $A_n$? Among other things, we give a positive answer for any $m$ having any divisor $d$ in the range $3 \leq d \leq o(n^{1/2})$.

math.CO↗
arXiv2019

Random Generation of the Special Linear Group

It is well known that the proportion of pairs of elements of $\operatorname{SL}(n,q)$ which generate the group tends to $1$ as $q^n\to \infty$. This was proved by Kantor and Lubotzky using the classification of finite simple groups. We give a proof of this theorem which does not depend on the classification. An essential step in our proof is an estimate for the average of $1/\operatorname{ord} g$ when $g$ ranges over $\operatorname{GL}(n,q)$, which may be of independent interest. We prove that this average is \[ \exp(-(2-o(1)) \sqrt{n \log n \log q}). \]

math.GR↗
arXiv2019

Random generation under the Ewens distribution

The Ewens sampling formula with parameter $α$ is the distribution on $S_n$ which gives each $π\in S_n$ weight proportional to $α^{C(π)}$, where $C(π)$ is the number of cycles of $π$. We show that, for any fixed $α$, two Ewens-random permutations generate at least $A_n$ with high probability. More generally we work out how many permutations are needed for $α$ growing with $n$. Roughly speaking, two are needed for $0 \leq α\ll n^{1/2}$, three for $n^{1/2} \ll α\ll n^{2/3}$, etc.

math.GR↗
arXiv2017

The Probability of Generating the Symmetric Group

We consider the probability $p(S_n)$ that a pair of random permutations generates either the alternating group $A_n$ or the symmetric group $S_n$. Dixon (1969) proved that $p(S_n)$ approaches $1$ as $n\to\infty$ and conjectured that $p(S_n)=1-1/n+o(1/n)$. This conjecture was verified by Babai (1989), using the Classification of Finite Simple Groups. We give an elementary proof of this result; specifically we show that $p(S_n)=1-1/n+\mathcal {O}(n^{-2+ε})$. Our proof is based on character theory and character estimates, including recent work by Schlage-Puchta (2012).

math.GR↗
arXiv2017

The trivial lower bound for the girth of $S_n$

Consider the Cayley graph of $S_n$ generated by a random pair of elements $x,y$. Conjecturally, the girth of this graph is $Ω(n \log n)$ with probability tending to $1$ as $n\to\infty$. We show that it is at least $Ω(n^{1/3})$.

math.GR↗
arXiv2017

An improved lower bound for Folkman's theorem

Folkman's Theorem asserts that for each $k \in \mathbb{N}$, there exists a natural number $n = F(k)$ such that whenever the elements of $[n]$ are two-coloured, there exists a set $A \subset [n]$ of size $k$ with the property that all the sums of the form $\sum_{x \in B} x$, where $B$ is a nonempty subset of $A$, are contained in $[n]$ and have the same colour. In 1989, Erdős and Spencer showed that $F(k) \ge 2^{ck^2/ \log k}$, where $c >0$ is an absolute constant; here, we improve this bound significantly by showing that $F(k) \ge 2^{2^{k-1}/k}$ for all $k\in \mathbb{N}$.

math.CO↗