TY - JOUR
T1 - Cohomology-developed matrices
T2 - constructing families of weighing matrices and automorphism actions
AU - Goldberger, Assaf
AU - Dula, Giora
N1 - Publisher Copyright:
© The Author(s) 2024.
PY - 2024/11
Y1 - 2024/11
N2 - The aim of this work is to construct families of weighing matrices via their automorphism group action. The matrices can be reconstructed from the 0, 1, 2-cohomology groups of the underlying automorphism group. We use this mechanism to (re)construct the matrices out of abstract group datum. As a consequence, some old and new families of weighing matrices are constructed. These include the Paley conference, the projective space, the Grassmannian, and the flag variety weighing matrices. We develop a general theory relying on low-dimensional group cohomology for constructing automorphism group actions and in turn obtain structured matrices that we call cohomology-developed matrices. This ‘cohomology development’ generalizes the cocyclic and group developments. The algebraic structure of modules of cohomology-developed matrices is discussed, and an orthogonality result is deduced. We also use this algebraic structure to define the notion of quasiproducts, which is a generalization of the Kronecker product.
AB - The aim of this work is to construct families of weighing matrices via their automorphism group action. The matrices can be reconstructed from the 0, 1, 2-cohomology groups of the underlying automorphism group. We use this mechanism to (re)construct the matrices out of abstract group datum. As a consequence, some old and new families of weighing matrices are constructed. These include the Paley conference, the projective space, the Grassmannian, and the flag variety weighing matrices. We develop a general theory relying on low-dimensional group cohomology for constructing automorphism group actions and in turn obtain structured matrices that we call cohomology-developed matrices. This ‘cohomology development’ generalizes the cocyclic and group developments. The algebraic structure of modules of cohomology-developed matrices is discussed, and an orthogonality result is deduced. We also use this algebraic structure to define the notion of quasiproducts, which is a generalization of the Kronecker product.
KW - Cocyclic development
KW - Group cohomology
KW - Weighing matrices
UR - http://www.scopus.com/inward/record.url?scp=85184604953&partnerID=8YFLogxK
U2 - 10.1007/s10801-024-01346-7
DO - 10.1007/s10801-024-01346-7
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AN - SCOPUS:85184604953
SN - 0925-9899
VL - 60
SP - 603
EP - 665
JO - Journal of Algebraic Combinatorics
JF - Journal of Algebraic Combinatorics
IS - 3
ER -