DisCoMath Seminar: Cubes of Symmetric Designs and Group Actions
DisCoMath Seminar
Cubes of Symmetric Designs and Group Actions
Kristijan Tabak
Professional studies professor
RIT Croatia
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Abstract:
A 3-dimensional cube of a symmetric (v, k, λ) design is a map from [v]^3 to {0, 1} such that every slice of a cube is an incidence matrix of a (v, k, λ) symmetric design. We analyze a regular action of a group G of order v on slices of a cube. We generalize classical results about regular action of a group on a symmetric design to a case of a dimension 3. Additionally, we prove that a group action on one set of parallel slices fully determines an action of G on other perpendicular slices. We also prove that it is not possible to have parallel slices as left and right G-orbits simultaneously. Furthermore, we prove that possible G orbits of mutually perpendicular slices alternate in a sense of being a left and right G-orbit.
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