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Computations of Colorings 7D-Hypercube's Hyperplanes for All Irreducible Representations.

Krishnan Balasubramanian1

  • 1School of Molecular Sciences, Arizona State University, Tempe, Arizona.

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|December 24, 2019
PubMed
Summary

This study enumerates colorings of the 7D-hypercube and its hyperplanes using 110 irreducible representations (IRs). These computations are vital for understanding complex systems in chemistry and biology.

Keywords:
7D-hyper octahedral group7D-hypercubechemical and biological applicationscomputation of hyperplane colorings of 7D-hypercubeswreath product character cycle indices for hyperplanes of 7D-cube

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Area of Science:

  • Combinatorics
  • Group Theory
  • Computational Chemistry
  • Computational Biology

Background:

  • The 7D-hypercube and its hyperplanes are relevant to diverse fields including chemistry and biology.
  • Applications include modeling the periodic table, water clusters, genetic networks, and large datasets.
  • Understanding symmetry and coloring is crucial for these applications.

Purpose of the Study:

  • To compute and enumerate the colorings of the 7D-hypercube for all hyperplanes (q=1-7).
  • To analyze these colorings across all 110 irreducible representations (IRs) of the seventh-dimensional hyperoctahedral group.
  • To provide explicit computational tables for these colorings.

Main Methods:

  • Utilized the Möbius inversion technique.
  • Employed generalized character cycle indices for 110 IRs.
  • Calculated generating functions for hypercube colorings.

Main Results:

  • Enumerated colorings for all seven types of hyperplanes of the 7D-hypercube.
  • Results are presented for all 110 IRs of the hyperoctahedral group.
  • Provided explicit tables detailing the computed colorings for q=1 to q=7.

Conclusions:

  • The study provides a comprehensive enumeration of 7D-hypercube colorings across various symmetry representations.
  • The methods and results offer valuable tools for applications in chemistry, biology, and data science.
  • The generated tables serve as a foundational resource for further research in related fields.