15 papers
Complex generalised weighing matrices in centraliser algebras of monomial representations
Ronan Egan, Padraig Ó Catháin, Andrea Švob
An matrix with exactly non-zero entries taken from the set of complex roots of unity in each row and column satisfying is a com…
Cliques in graphs constructed from Strongly Orthogonal Subsets in exceptional root systems
Patrick J. Browne, Pádraig Ó Catháin
Given a root system , two roots are said to be \emph{strongly orthogonal} if neither their sum nor difference is a root. Gashi defined a family of graphs with vertices labelled…
Symmetries of regular -graphs
Daniel R Hawtin, Padraig Ó Catháin
Given a finite vector space , the -analogue of a graph, called a -graph, is a pair , where is the set of -dime…
Extendibility of Latin Hypercuboids
Candida Bowtell, Alice Devillers, André Kündgen +2
A Latin hypercuboid of order is a -dimensional matrix of dimensions , with symbols from a set of cardinality such that each symbol…
A generalisation of bent vectors for Butson Hadamard matrices
José Andrés Armario, Ronan Egan, Hadi Kharaghani +1
An complex matrix with entries in the roots of unity which satisfies is called a Butson Hadamard matrix. While a matrix with…
Centraliser algebras of monomial representations and applications in combinatorics
Santiago Barrera Acevedo, Padraig Ó Catháin, Heiko Dietrich +1
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…
Triangle-free graphs with diameter 2
Alice Devillers, Nina Kamčev, Brendan McKay +5
There are finitely many graphs with diameter and girth 5. What if the girth 5 assumption is relaxed? Apart from stars, are there finitely many triangle-free graphs with diamete…
A Bollobás-type problem: from root systems to Erdős-Ko-Rado
Patrick J. Browne, Qëndrim R. Gashi, Padraig Ó Catháin
Motivated by an Erdős--Ko--Rado type problem on sets of strongly orthogonal roots in the root system, we estimate bounds for the size of a family of pairs $(A_{i}, B_{i}…
Segre's theorem on ovals in Desarguesian projective planes
Patrick J. Browne, Steven T. Dougherty, Padraig Ó Catháin
Segre's theorem on ovals in projective spaces is an ingenious result from the mid-twentieth century which requires surprisingly little background to prove. This note, suitable for…
Morphisms of Skew Hadamard Matrices
Philip Heikoop, Guillermo Nuñez Ponasso, Padraig Ó Catháin +1
Quaternary unit Hadamard (QUH) matrices were introduced by Fender, Kharagani and Suda along with a method to construct them at prime power orders. We present a novel construction o…
Dimensions of semi-simple matrix algebras
Phillip Heikoop, Padraig Ó Catháin
For we show that every integer of the form such that is the dimension of a connected semi-simple subalgebra of…
Morphisms of Butson classes
Ronan Egan, Padraig Ó Catháin
We introduce the concept of a morphism from the set of Butson Hadamard matrices over kth roots of unity to the set of Butson matrices over th roots of unity. As concrete exam…
Classifying cocyclic Butson Hadamard matrices
Ronan Egan, Dane Flannery, Padraig Ó Catháin
We classify all the cocyclic Butson Hadamard matrices of order over the th roots of unity for an odd prime and . That is, we compile a lis…
Trades in complex Hadamard matrices
Padraig Ó Catháin, Ian M. Wanless
A trade in a complex Hadamard matrix is a set of entries which can be changed to obtain a different complex Hadamard matrix. We show that in a real Hadamard matrix of order all…
An asymptotic existence result on compressed sensing matrices
Darryn Bryant, Padraig Ó Catháin
For any rational number and all sufficiently large we give a deterministic construction for an compressed sensing matrix with -recov…