Doubly transitive lines I: Higman pairs and roux

TL;DR

This paper explores the structure of doubly transitive lines in complex vector spaces, revealing their optimal packing properties and introducing a new generalization related to algebraic combinatorics.

Contribution

It establishes the optimality of doubly transitive lines in complex projective space and introduces a novel generalization of certain algebraic combinatorial structures.

Findings

Doubly transitive lines are optimal packings in complex projective space.

Connection established between doubly transitive lines and algebraic combinatorics.

Introduction of a new generalization of regular abelian distance-regular antipodal covers.

Abstract

We study lines through the origin of finite-dimensional complex vector spaces that enjoy a doubly transitive automorphism group. In doing so, we make fundamental connections with both discrete geometry and algebraic combinatorics. In particular, we show that doubly transitive lines are necessarily optimal packings in complex projective space, and we introduce a fruitful generalization of regular abelian distance-regular antipodal covers of the complete graph.