Abstract
We consider the graph $E_{3,1}$ with three generators $\sigma_1, \sigma_2, \delta$, where $\sigma_1$ has an edge with each of $\;\sigma_2$ and $\;\delta$. We then define the Artin group of the graph $E_{3,1}$ and consider its reduced Perron representation of degree three. After we specialize the indeterminates used in defining the representation to non-zero complex numbers, we obtain a necessary and sufficient condition that guarantees the irreducibility of the representation.<br />
Highlights
To any undirected simple graph T, we introduce the Artin group, A, which is defined as an abstract group with vertices of Γ as its generators and two relations: xy = yx for vertices x and y that have no edge in common and xyx = yxy if the vertices x and y have a common edge.Let An be the graph having n vertices σi’s (1 ≤ i ≤ n ) in which σi and σi+1 share a comon edge, where i = 1, 2, ..., n − 1
After we specialize the indeterminates used in defining the representation to non-zero complex numbers, we obtain a necessary and sufficient condition that guarantees the irreducibility of the representation
We notice that the Artin group of An is the braid group on n + 1 strands
Summary
To any undirected simple graph T , we introduce the Artin group, A, which is defined as an abstract group with vertices of Γ as its generators and two relations: xy = yx for vertices x and y that have no edge in common and xyx = yxy if the vertices x and y have a common edge. Having defined An, we consider En+1,p, which is the graph obtained from An by adding a vertex δ and an edge connecting σp and δ. Knowing the reduced Burau representation of Bn+1 of degree n, Perron extends such a representation to a representation of Bn+1 of degree 2n. The representation obtained is referred to as Burau bis representation. Dn to non zero complex numbers, and we study this representation explicitly in the case n = 2 and p = 1. We reduce the complex specialization of the representation ψλ to a representation of degree 3, namely A(E3,1) → GL3(C). A necessary and sufficient condition which guarantees its irreducibility is obtained in that case
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