Centraliser algebras of monomial representations and applications in combinatorics
Santiago Barrera Acevedo, Padraig Ó Catháin, Heiko Dietrich, Ronan Egan
TL;DR
The paper develops a cohesive framework for centraliser algebras of finite, monomial group representations and applies it to construct and classify complex Hadamard matrices with high symmetry. It combines representation theory, group actions, and computational algebra (notably Gröbner bases) to produce an explicit basis for centraliser algebras, express eigenvalues as character sums, and solve norm equations that determine Hadamard properties. A key contribution is locating group-developed and cocyclic Hadamard matrices within centralisers: group-developed matrices correspond to centralisers of the right regular representation, while cocyclic matrices arise from central extensions via 2-cocycles and monomial covers. The methodology yields practical algorithms for constructing Hadamard matrices with prescribed symmetries and provides computational classifications for primitive groups of small degree, with Paley-type and other known families appearing naturally. Beyond Hadamard matrices, the approach extends to weighing matrices and related combinatorial structures, offering a versatile tool for algebraic combinatorics and design theory, with potential implications for quantum information and signal processing.
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 Gröbner bases to produce a new classification of highly symmetric complex Hadamard matrices.