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An Atomic Model of the Graph of Permutations Linked by Transpositions by Achim Kehrein; Frank Zimmer; Nicolas Fischöder is a Social Sciences article available to read on EtoBox.

What is An Atomic Model of the Graph of Permutations Linked by Transpositions about?

This article presents a new perspective on the structure of the symmetric group. We consider the graph whose vertices are the permutations. Two permutations are linked when one is the product of a transposition and the other. We construct three-dimensional models of this graph mimicking atomic models with their shells and orbitals. The kth shell consists of the permutations which can be factored into k transpositions but not fewer. We distribute the vertices of a shell on a sphere as symmetrically as possible. Higher shells are placed on spheres with larger radii. We also use symmetry to determine the relative placement of shells. Further, we split shells into orbitals corresponding to the cycle types of the permutations. Orbitals belonging to the same shell are placed on the same sphere but with independent symmetry constellations. Overall, the constructed graphs provide a new visual approach to the symmetric group.

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Author

Achim Kehrein; Frank Zimmer; Nicolas Fischöder

Publisher

Walter de Gruyter GmbH

Published

2022

Language

EN

Field

Social Sciences