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We introduce pattern geometries, a new class of coset geometries constructed from pattern groups inside unitriangular groups \( U_n(q) \). Using a coset geometry framework, we define these geometries via families of maximal parabolic subgroups derived from the covering relations of an underlying poset.
We show that these geometries are flag-transitive, residually connected, and firm, and we characterize when they are thin or thick. For \( q=2 \), they yield an infinite family of regular hypertopes, providing new highly symmetric combinatorial structures.
Our work connects the combinatorics of pattern groups with incidence geometry and extends the study of coset geometries for Sylow subgroups of classical groups. These constructions provide new examples of abstract polytopes and hypertopes, contributing to the broader understanding of symmetry and independence properties in group-generated geometries.
This is joint work with C. Piedade.
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