arXiv subjects

Charles Leedham-Green

Publications and source records attributed to Charles Leedham-Green.

4 recordsLinked to original sources

arXiv2023

Condorcet Domains of Degree at most Seven

In this paper we give the first explicit enumeration of all maximal Condorcet domains on $n\leq 7$ alternatives. This has been accomplished by developing a new algorithm for constructing Condorcet domains, and an implementation of that algorithm which has been run on a supercomputer. We follow this up by the first survey of the properties of all maximal Condorcet domains up to degree 7, with respect to many properties studied in the social sciences and mathematical literature. We resolve several open questions posed by other authors, both by examples from our data and theorems. We give a new set of results on the symmetry properties of Condorcet domains which unify earlier works. Finally we discuss connections to other domain types such as non-dictatorial domains and generalisations of single-peaked domains. All our data is made freely available for other researches via a new website.

cs.DM↗

arXiv2023

The Largest Condorcet Domains on 8 Alternatives

In this note, we report on a record-breaking Condorcet domain (CD) for n=8 alternatives. We show that there exists a CD of size 224, which is optimal and essentially unique (up to isomorphism). If we consider the underlying permutations and focus on Condorcet domains containing the identity permutation, 56 isomorphic such Condorcet domains exist. Our work sheds light on the structure of CDs and UCDs and has potential applications in voting theory and social choice.

math.CO↗

arXiv2000

Explicit computation of Galois p-groups unramified at p

In this paper we introduce a new method for finding Galois groups by computer. This is particularly effective in the case of Galois groups of p-extensions ramified at finitely many primes but unramified at the primes above p. Such Galois groups have been regarded as amongst the most mysterious objects in number theory. Very little has hitherto been discovered regarding them despite their importance in studying p-adic Galois representations unramified at p. The conjectures of Fontaine-Mazur say that they should have no p-adic analytic quotients (equivalently, the images of the Galois representations should always be finite), and there are generalizations due to the first author, suggesting that they should instead have `large' actions on certain trees. The idea is to modify the p-group generation algorithm so that as it goes along, it uses number-theoretical information to eliminate groups that cannot arise as suitable quotients of the Galois group under investigation. In the best cases we obtain a short list of candidates for the Galois group. This leads to various conjectures.

math.NT↗

arXiv1998

Counterexamples to a conjecture of Lemmermeyer

We produce infinitely many finite 2-groups that do not embed with index 2 in any group generated by involutions. This disproves a conjecture of Lemmermeyer and restricts the possible Galois groups of unramified 2-extensions, Galois over the rationals, of quadratic number fields.

math.NT↗