On the Irreducibility of Perron Representations of Degrees 4 and 5

Authors

  • Malak M. Dally Department of Mathematics Beirut Arab University P.O. Box: 11-5020 Beirut, Lebanon
  • Mohammad N. Abdulrahim Department of Mathematics Professor Beirut Arab University P.O. Box: 11-5020 Beirut, Lebanon

DOI:

https://doi.org/10.29020/nybg.ejpam.v11i1.3199

Keywords:

Artin representation, braid group, Burau representation, graph, irreducibility

Abstract

We consider the graph $E_{n+1,1}$ with (n+1) generators $\sigma_1,..., \sigma_{n}$, and $\delta$, where $\sigma_{i}$ has an edge with $\sigma_{i+1}$ for $i=1,...,n+1$, and $
\sigma_{1}$ has an edge with $\delta$. We then define the Artin group of the graph $E_{n+1,1}$ for $n=3$ and $n=4$ and consider its reduced Perron's representation of degrees
four and five respectively. After we specialize the indeterminates used in defining the representation to non-zero complex numbers, we obtain necessary and sufficient
conditions that guarantee the irreducibility of the representations for $n=3$ and $4$ .

Author Biographies

  • Malak M. Dally, Department of Mathematics Beirut Arab University P.O. Box: 11-5020 Beirut, Lebanon

    Department of Mathematics

    Ph.D. student

  • Mohammad N. Abdulrahim, Department of Mathematics Professor Beirut Arab University P.O. Box: 11-5020 Beirut, Lebanon

    Department of Mathematics

    Professor

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Published

2018-01-30

Issue

Section

Mathematical Biosciences

How to Cite

On the Irreducibility of Perron Representations of Degrees 4 and 5. (2018). European Journal of Pure and Applied Mathematics, 11(1), 215-237. https://doi.org/10.29020/nybg.ejpam.v11i1.3199