arXiv subjects

Santiago Barrera Acevedo

Publications and source records attributed to Santiago Barrera Acevedo.

4 recordsLinked to original sources

arXiv2026

Computing canonical labellings of finite solvable groups

We define a canonical labelling function on the class of finite solvable groups so that two such groups $G$ and $H$ are isomorphic if and only if can$(G)=$can$(H)$. Specifically, can$(G)$ is a group presentation that describes a group isomorphic to $G$, and our description explains how to construct an isomorphism $G\to$can$(G)$. Our approach is motivated by O'Brien's (1993) canonical presentations for finite $p$-groups and utilises ideas from group cohomology first described by Robinson (1982) and automorphism group algorithms developed by Smith (1994), Holt (2001), and others. We also discuss a proof-of-concept implementation for the computer algebra system GAP and comment on the major bottlenecks and open research questions.

math.GR↗

arXiv2026

Automorphisms of Kimura Hadamard Matrices

We investigate the structure of the automorphism groups of Kimura Hadamard matrices (KHMs) constructed from dihedral groups. We identify several different types of automorphisms, and show that the automorphism group of a KHM always has a subgroup isomorphic to $D_{2k}\times Q_8$, or $C_2\times D_{2k}\times Q_8$ if it is $y$-invariant. We exhibit additional automorphisms arising from the holomorph of the dihedral group under suitable structural conditions. A comparison with known examples, including those of Kimura, Niwasaki, and matrices arising from the Shinoda--Yamada construction, reveals counterexamples to a conjecture of Ó Catha\'ın and suggests that no further automorphisms occur beyond those predicted by our framework.

math.CO↗

arXiv2025

Centraliser algebras of monomial representations and applications in combinatorics

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.

math.CO↗

arXiv2019

Constructing cocyclic Hadamard matrices of order 4p

Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as a class of Hadamard matrices with interesting algebraic properties. Ó Catháin and Röder described a classification algorithm for CHMs of order $4n$ based on relative difference sets in groups of order $8n$; this led to the classification of all CHMs of order at most 36. Based on work of de Launey and Flannery, we describe a classification algorithm for CHMs of order $4p$ with $p$ a prime; we prove refined structure results and provide a classification for $p \leqslant 13$. Our analysis shows that every CHM of order $4p$ with $p\equiv 1\bmod 4$ is equivalent to a Hadamard matrix with one of five distinct block structures, including Williamson type and (transposed) Ito matrices. If $p\equiv 3 \bmod 4$, then every CHM of order $4p$ is equivalent to a Williamson type or (transposed) Ito matrix.

math.CO↗