Centraliser algebras of monomial representations and applications in combinatorics

TL;DR

This paper constructs explicit bases for centraliser algebras of monomial group representations, linking them to combinatorial objects like Hadamard matrices and applying Gr"obner bases for classification.

Contribution

It provides a new explicit basis construction for these algebras and connects them to Hadamard matrices and their classification.

Findings

Explicit basis for centraliser algebra of monomial representations

Character table constructed via character sums over double cosets

New classification of symmetric complex Hadamard matrices using Gr"obner bases

Abstract

Centraliser algebras of monomial representations of finite groups may be constructed and studied using methods similar to those employed in the study of permutation groups. Guided by results of D. G. Higman and others, we give an explicit construction for a basis of the centraliser algebra of a monomial representation. The character table of this algebra is then constructed via character sums over double cosets. We locate the theory of group-developed and cocyclic-developed Hadamard matrices within this framework. We apply Gr\"obner bases to produce a new classification of highly symmetric complex Hadamard matrices.