On the irreducibility of local representations of the Braid group $B_n$

Abstract

We prove that any homogeneous local representation $\varphi :B_n \rightarrow GL_n(\mathbb {C})$ of type 1 or 2 of dimension $n\ge 6$ is reducible. Then, we prove that any representation $\varphi :B_n \rightarrow GL_n(\mathbb {C})$ of type 3 is equivalent to a complex specialization of the standard representation $\tau _n$. Also, we study the irreducibility of all local linear representations of the braid group $B_3$ of degree 3. We prove that any local representation of type 1 of $B_3$ is reducible to a Burau type representation and that any local representation of type 2 of $B_3$ is equivalent to a complex specialization of the standard representation. Moreover, we construct a representation of $B_3$ of degree 6 using the tensor product of local representations of type 2. Let $u_i$, $i=1,2$, be non-zero complex numbers on the unit circle. We determine a necessary and sufficient condition that guarantees the irreducibility of the obtained representation.

1 Introduction

The braid group $B_n$ is represented, due to Artin, in the group Aut($F_n$) of automorphisms of the free group $F_n$ generated by $x_1, \ldots , x_n$. To attack the linearity of the braid group $B_n$, the faithfulness of the braid group representations was studied. One of these represenations is the Burau representation which was, for a long time, a candidate to answer the question of faithfulness of the braid group $B_n$. It was proved that the Burau representation is faithful for $n\le 3$ and not faithful for $n\ge 5$. For $n=4$, the question of faithfulness of the Burau representation has not been answered yet. For more details, see [2] and [3].

In addition to the linearity of the braid group $B_n$, the classification of irreducible complex representations of $B_n$ was of great concern. In [4], Formanek found a necessary and sufficient condition for the specialization of the reduced Burau representation to be irreducible. Moreover, Formanek classified all irreducible complex representations of the braid group $B_n$ of degree at most $n-1$ for $n\ge 7$. In [7], Sysoeva extended this classification to representations of degree n for $n\ge 9$. For $n=5, 6, 7$ and 8, the classification was completed by Formanek, Lee, Sysoeva and Vazirani. For more details, see [5]. For $n \ge 10$, Sysoeva proved, in [8], that there are no irreducible representations of $B_n$ of dimension $n+1$.

The local representations of the braid group $B_3$ were studied by Mikhalchishina who proved that any local representation of the braid group $B_3$ into $GL_3(\mathbb {C})$ is of type 1 or 2. In addition, Mikhalchishina studied the n-dimensional homogeneous local representations $\varphi$ of the braid group $B_n$ and proved that $\varphi$ coincides with one of the three representations $\varphi _1$, $\varphi _2$ and $\varphi _3$ which were defined. For more details, see [6].

In our work, first we study the irreducibility of the local homogeneous multi-parameter representations of types 1 and 2 of degree n of the braid group $B_n$. We prove that, for $n\ge 6$, any homogeneous local representation of type 1 or 2 is reducible.

Next, we consider the case of homogeneous local representations of type 3 of degree n of the braid group $B_n$. We prove that any homogeneous local representation of type 3 is equivalent to a complex specialization of the standard representation. Consequently, any multi-parameter homogeneous local representation of type 3 is irreducible if and only if $bc\ne 1$.

Then, we study the irreducibility of all local representations of the braid group $B_3$. We prove that any three-dimensional local representation of type 1 is reducible to a representation of Burau type. Also, we prove that any three-dimensional local representation of type 2 is equivalent to a complex specialization of the standard representation. Due to this equivalence, any local representation of type 2 is irrreducible if and only if $bc\ne 1$.

Finally, we find the tensor product of two complex specializations of the standard representations of $B_3$. We prove that the obtained nine-dimensional multi-parameter representation is a direct sum of a complex specialization of the standard representation and a six-dimensional representation $\varphi$. We consider the case when the complex numbers $u_i's$ are on the unit circle. Then, we prove that $\varphi$ is irreducible if and only if $\sqrt{u_1} \ne \pm \sqrt{u_2}$.

2 Preliminaries

Definition 2.1

[1] The braid group, $B_n$, is an abstract group generated by $\sigma _1, \sigma _2, \ldots , \sigma _{n-1}$ with the following relations

$$\begin{aligned} \sigma _i\sigma _j=\sigma _j\sigma _i,\hbox { for all }i,j=1,\ldots ,n-1\hbox { with }|i-j|\ge 2, \end{aligned}$$

and

$$\begin{aligned} \sigma _i\sigma _{i+1}\sigma _i=\sigma _{i+1}\sigma _i\sigma _{i+1},\hbox { for }i=1,\ldots ,n-2. \end{aligned}$$

Definition 2.2

[7, Definition 2] The corank of the representation $\rho : B_n\rightarrow GL_r(\mathbb {C})$ is $rank(\rho (\sigma _i)-1)$, where the $\sigma _i$ are the standard generators of the group $B_n$.

Definition 2.3

[6] A representation $\varphi : B_n \longrightarrow GL_n(\mathbb {C})$ is called local if

$$\begin{aligned} \varphi (\sigma _i)= \left( \begin{array}{c|c|c} I_{i-1} &{} 0 &{} 0 \\ \hline 0 &{} R_i &{} 0 \\ \hline 0 &{} 0 &{} I_{n-i-1} \end{array} \right) , \quad i= 1, 2,\ldots , n-1, \end{aligned}$$

where $I_m$ is the identity matrix of order m and $R_i$ is a matrix of order 2. A local representation is called homogeneous if $R_1=R_2=\cdots =R_{n-1}$.

Theorem 2.4

[6, Theorem 1] If $\varphi : B_3 \rightarrow GL_3(\mathbb {C})$ is a local representation then $\varphi$ has one of the two types:

• (1) $\varphi (\sigma _1) = \left( \begin{array}{cc|cc} \alpha (1-d) &{} \frac{(1-d)(1-\alpha +d\alpha )}{c} &{} 0 \\ c &{} d &{} 0 \\ \hline 0 &{} 0 &{} 1 \\ \end{array} \right)$, $\varphi (\sigma _2) = \left( \begin{array}{c|cc} 1 &{} 0 &{} 0 \\ \hline 0 &{} \alpha &{} \frac{(1-\alpha )(1-d +d\alpha )}{\gamma } \\ 0 &{} \gamma &{} d(1-\alpha ) \\ \end{array} \right)$,

  where $d, \alpha \ne 1$ and $c, \gamma \ne 0$;

• (2) $\varphi (\sigma _1) = \left( \begin{array}{cc|cc} 0 &{} b &{} 0 \\ c &{} 0 &{} 0 \\ \hline 0 &{} 0 &{} 1 \\ \end{array} \right)$, $\varphi (\sigma _2) = \left( \begin{array}{c|cc} 1 &{} 0 &{} 0 \\ \hline 0 &{} 0 &{} \frac{bc}{\gamma } \\ 0 &{} \gamma &{} 0 \\ \end{array} \right)$, where $bc, \gamma \ne 0$.

Corollary 2.5

[6, Corollary to Theorem 1] If $\varphi : B_n \longrightarrow GL_n(\mathbb {C})$, $n\ge 3$, is a homogeneous local representation, then $\varphi$ coincides with one of the representations $\varphi _1$, $\varphi _2$ and $\varphi _3$ defined as follows:

$$\begin{aligned} \varphi _j: B_n \longrightarrow GL_n(\mathbb {C}), \end{aligned}$$

$$\begin{aligned} (1)\hspace{7.22743pt}\varphi _1(\sigma _i)= \left( \begin{array}{c|c|c} I_{i-1} &{} 0 \hspace{14.45377pt}0 &{} 0 \\ \hline 0 &{} \alpha \hspace{14.45377pt}\frac{1-\alpha }{\gamma } &{} 0 \\ 0 &{} \gamma \hspace{14.45377pt}0 &{} 0 \\ \hline 0 &{} 0 \hspace{14.45377pt}0 &{} I_{n-i-1} \end{array} \right) , \quad \gamma \ne 0, i= 1, 2,\ldots , n-1.\\ (2)\hspace{7.22743pt}\varphi _2(\sigma _i)= \left( \begin{array}{c|c|c} I_{i-1} &{} 0 \hspace{14.45377pt}0 &{} 0 \\ \hline 0 &{} 0 \hspace{14.45377pt}\frac{1-d}{c} &{} 0 \\ 0 &{} c \hspace{14.45377pt}d &{} 0 \\ \hline 0 &{} 0 \hspace{14.45377pt}0 &{} I_{n-i-1} \end{array} \right) , \quad c \ne 0, i= 1, 2,\ldots , n-1.\\ (3)\hspace{7.22743pt}\varphi _3(\sigma _i)= \left( \begin{array}{c|c|c} I_{i-1} &{} 0 \hspace{14.45377pt}0 &{} 0 \\ \hline 0 &{} 0 \hspace{14.45377pt}b &{} 0 \\ 0 &{} c \hspace{14.45377pt}0 &{} 0 \\ \hline 0 &{} 0 \hspace{14.45377pt}0 &{} I_{n-i-1} \end{array} \right) , \quad bc\ne 0, i= 1, 2,\ldots , n-1. \end{aligned}$$

Definition 2.6

[7, Definition 6] The standard representation is the representation

$$\begin{aligned} \tau _n: B_n \rightarrow GL_n(\mathbb {Z}[t^{\pm 1}]) \end{aligned}$$

defined by

$$\begin{aligned} \tau _n(\sigma _i)= \left( \begin{array}{c|c|c} I_{i-1} &{} 0 \hspace{14.45377pt}0 &{} 0 \\ \hline 0 &{} 0 \hspace{14.45377pt}t &{} 0 \\ 0 &{} 1 \hspace{14.45377pt}0 &{} 0 \\ \hline 0 &{} 0 \hspace{14.45377pt}0 &{} I_{n-i-1} \end{array} \right) , \end{aligned}$$

for $i=1,2, \ldots , n-1$, where $I_k$ is the $k \times k$ identity matrix.

Definition 2.7

[7] The complex specialization of the standard representation is defined by

$$\begin{aligned} \tau _n(u): B_n \rightarrow GL_n(\mathbb {C}), \end{aligned}$$

$$\begin{aligned} \tau _n(u)(\sigma _i)= \left( \begin{array}{c|c|c} I_{i-1} &{} 0 \hspace{14.45377pt}0 &{} 0 \\ \hline 0 &{} 0 \hspace{14.45377pt}u &{} 0 \\ 0 &{} 1 \hspace{14.45377pt}0 &{} 0 \\ \hline 0 &{} 0 \hspace{14.45377pt}0 &{} I_{n-i-1} \end{array} \right) , \end{aligned}$$

for $i=1,2,\ldots , n-1$, where $I_k$ is the $k \times k$ identity matrix, and u $\in \mathbb {C}^*$.

Lemma 2.8

[7, Lemma 5.3] If $u=1$, then $\tau _n(u)$ is reducible.

Lemma 2.9

[7, Lemma 5.4] If $u\ne 1$, then $\tau _n(u)$ is irreducible.

Theorem 2.10

[7, Theorem 5.5] Let $\rho : B_n:\rightarrow GL_r(\mathbb {C})$ be an irreducible representation of $B_n$ for $n\ge 6$. Let $r\ge n$, and let $\rho (\sigma _1)=1+A_1$ with $rank(A_1)=2$. Then $r=n$ and $\rho$ is equivalent to the representation $\tau _n(u)$, where u $\in \mathbb {C}^*$ and $u \ne 1$.

Definition 2.11

[4] The complex specialization of the reduced Burau representation $\beta _3(z)$ is defined by:

$$\begin{aligned} \beta _3(z): B_3 \rightarrow GL_2(\mathbb {C}), \end{aligned}$$

$$\begin{aligned} \begin{aligned} \beta _3(z)(\sigma _1) = \left( \begin{array}{cc} -z &{}{} 0 \\ -1 &{}{} 1 \\ \end{array} \right) \text{ and } \beta _3(z)(\sigma _2) = \left( \begin{array}{cc} 1 &{}{} -z \\ 0 &{}{} -z \\ \end{array} \right) . \end{aligned} \end{aligned}$$

Theorem 2.12

[4, Theorem 11] Let $\rho :B_3 \rightarrow GL_2(\mathbb {C})$ be an irreducible representation. Then $\rho$ is equivalent to $\chi (y)\otimes \beta _3(z)$ for some $y, z \in \mathbb {C}^*$, where z is not a root of the polynomial $t^2+t+1$. Here $\chi (y)$ is the one dimentional representation and $\beta _3(z)$ is the reduced Burau representation. We say that $\rho$ is of Burau type.

Definition 2.13

The principal square root function is the function defined as follows: For all $z\in \mathbb {C}, z=(\rho , \alpha ), \rho \ge 0$. $\sqrt{z}=\sqrt{\rho }e^{i\frac{\alpha }{2}}$, where $-\pi <\alpha \le \pi .$

3 Irreduciblility of homogeneous local representations of $B_n$ for $n\ge 6$

Mikhalchishina proved, in [6, Proposition, p. 672], that type 1 and type 2 representations are not equivalent when $d\ne \alpha$. In this section, we prove that any homogeneous local representation of type 1 or 2 of dimension $n\ge 6$ is reducible. Then, we prove that any homogeneous local representation of type 3 is equivalent to a complex specialization of the standard representation $\tau _n$.

Theorem 3.1

The homogeneous local representations of types 1 and 2 are reducible for $n\ge 6$.

Proof

Let $\varphi _1$ and $\varphi _2$ be two homogeneous local representations of $B_n$ of types 1 and 2 respectiveley with $n\ge 6$. Consider the matrices $P_1$ and $P_2$ defined as

$$\begin{aligned} P_1=Diag\left( \frac{1}{\gamma ^{n-1}}, \cdots , \frac{1}{\gamma }, 1\right) \hbox { and } P_2=Diag\left( \frac{1}{c^{n-1}}, \frac{1}{c^{n-2}}, \cdots , \frac{1}{c}, 1\right) , \end{aligned}$$

where $Diag(a_1, a_2, \ldots , a_n)$ is a diagonal $n\times n$ matrix, with $a_{ii}=a_i$.

By direct computations, we get

$$\begin{aligned} P_1^{-1}\varphi _1(\sigma _i)P_1=\tilde{\varphi _1}(\sigma _i) \hbox { and } P_2^{-1}\varphi _2(\sigma _i)P_2=\tilde{\varphi _2}(\sigma _i) \end{aligned}$$

where,

$$\begin{aligned} \tilde{\varphi _1}(\sigma _i)= \left( \begin{array}{c|c|c} I_{i-1} &{} 0 \hspace{28.90755pt}0 &{} 0 \\ \hline 0 &{} \alpha \hspace{14.45377pt}1-\alpha &{} 0 \\ 0 &{} 1 \hspace{28.90755pt}0 &{} 0 \\ \hline 0 &{} 0 \hspace{28.90755pt}0 &{} I_{n-i-1} \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} \tilde{\varphi _2}(\sigma _i)= \left( \begin{array}{c|c|c} I_{i-1} &{} 0 \hspace{28.90755pt}0 &{} 0 \\ \hline 0 &{} 0 \hspace{14.45377pt}1-d &{} 0 \\ 0 &{} 1 \hspace{28.90755pt}d &{} 0 \\ \hline 0 &{} 0 \hspace{28.90755pt}0 &{} I_{n-i-1} \end{array} \right) \end{aligned}$$

for $i=1, 2, \ldots , n-1$.

This implies that the representations $\varphi _1$ and $\varphi _2$ are equivalent to the representations $\tilde{\varphi _1}$ and $\tilde{\varphi _2}$ respectively. Thus, we can verify that the corank of the representations $\varphi _1$ and $\varphi _2$ is 1. This implies that the representations $\varphi _1$ and $\varphi _2$ are reducible. (See [4], Theorem 10) $\square$

Now, we prove that any representation $\varphi _3:B_n \rightarrow GL_n(\mathbb {C})$ of type 3 is equivalent to a complex specialization of the standard representation $\tau _n$.

Theorem 3.2

Let $\varphi _3:B_n \rightarrow GL_n(\mathbb {C})$ be a homogeneous local representation of type 3. Then, the representation $\varphi _3$ is equivalent to a complex specialization the standard representation $\tau _n$.

Proof

Let $\varphi _3:B_n \rightarrow GL_n(\mathbb {C})$ be a homogeneous local representation of type 3. Consider the matrix $P$ defined by

$$\begin{aligned} P=Diag\left( \frac{1}{c^{n-1}}, \frac{1}{c^{n-2}}, \ldots , \frac{1}{c}, 1\right) , \end{aligned}$$

where $Diag(a_1, a_2, \ldots , a_n)$ is a diagonal $n\times n$ matrix with $a_{ii}=a_i$.

Direct computations show that

$$\begin{aligned} P^{-1}\sigma _iP =\left( \begin{array}{c|c|c} I_{i-1} &{} 0 \hspace{14.45377pt}0 &{} 0 \\ \hline 0 &{} 0 \hspace{14.45377pt}bc &{} 0 \\ 0 &{} 1 \hspace{14.45377pt}0 &{} 0 \\ \hline 0 &{} 0 \hspace{14.45377pt}0 &{} I_{n-i-1} \end{array} \right) . \end{aligned}$$

By letting $u=bc$, we find that this representation is equivalent to a complex specialization of the standard representation $\tau _n$.

By Lemmas 2.8 and 2.9, the standard representation is irreducible if and only if $u\ne 1$. This implies that $\varphi _3$ is irreducible if and only if $bc\ne 1$. $\square$

Thus, we state the following theorem:

Theorem 3.3

Let $\varphi _3:B_n \rightarrow GL_n(\mathbb {C})$ be a homogeneous local representation of type 3. Then, the representation $\varphi _3$ is irreducible if and only if $bc\ne 1$.

4 Irreducibility of local representations of $B_3$

In this section, we consider all the local representations of $B_3$. We prove that any local representation of type 1 of $B_3$ is reducible to a Burau type representation. Then, we prove that any local representation of type 2 of $B_3$ is equivalent to a complex specialization of the standard representation.

Theorem 4.1

Let $\varphi : B_3 \rightarrow GL_3(\mathbb {C})$ be a local representation of type 1. Then, $\varphi$ is reducible to a representation of Burau type.

Proof

Let $\varphi : B_3 \rightarrow GL_3(\mathbb {C})$ be a representation of type 1, then

$$\begin{aligned} \varphi (\sigma _1) = \left( \begin{array}{cc|cc} \alpha (1-d) &{} \frac{(1-d)(1-\alpha +d\alpha )}{c} &{} 0 \\ c &{} d &{} 0 \\ \hline 0 &{} 0 &{} 1 \\ \end{array} \right) , \varphi (\sigma _2) = \left( \begin{array}{c|cc} 1 &{} 0 &{} 0 \\ \hline 0 &{} \alpha &{} \frac{(1-\alpha )(1-d +d\alpha )}{\gamma } \\ 0 &{} \gamma &{} d(1-\alpha ) \\ \end{array} \right) , \end{aligned}$$

where $d, \alpha \ne 1$ and $c, \gamma \ne 0$.

We scale the basis using the matrix

$$\begin{aligned} P=Diag\left( \frac{1}{c}, 1, \gamma \right) . \end{aligned}$$

Thus, we get

$$\begin{aligned} P^{-1}\varphi (\sigma _1)P= \left( \begin{array}{ccc} \alpha (1-d) &{} (1-d)(1-\alpha +\alpha d) &{} 0 \\ 1 &{} d &{} 0 \\ 0 &{} 0&{} 1 \\ \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} P^{-1}\varphi (\sigma _2)P= \left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} \alpha &{} (1-\alpha )(1-d+\alpha d) \\ 0 &{} 1&{} (1-\alpha )d\\ \end{array} \right) . \end{aligned}$$

We have the following two cases:

Case 1. $\alpha (1-d)=1$. In this case, we have

$$\begin{aligned} \varphi (\sigma _1) = \left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 1 &{} d &{} 0 \\ 0 &{} 0 &{} 1 \\ \end{array} \right) \hbox { and } \varphi (\sigma _2) = \left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} \frac{1}{1-d} &{} \frac{-d(1-d+d^2)}{(-1+d)^2} \\ 0 &{} 1 &{} \frac{d^2}{-1+d} \\ \end{array} \right) . \end{aligned}$$

It is clear that the proper subspace $S=<e_2, e_3>$ is invariant. Consequently, the above representation $\varphi$ is reducible.

Restricting $\varphi$ to $S$, we obtain:

$$\begin{aligned} \varphi '(\sigma _1) = \left( \begin{array}{cc} d &{} 0 \\ 0 &{} 1 \\ \end{array} \right) \hbox { and } \varphi '(\sigma _2) = \left( \begin{array}{cc} \frac{1}{1-d} &{} \frac{-d(1-d+d^2)}{(-1+d)^2} \\ 1 &{} \frac{d^2}{-1+d} \\ \end{array} \right) . \end{aligned}$$

This formula is well defined since $d-1\ne 0$.

By direct computations, there is no proper invariant subspace of dimension one if $d \ne 0$ and $d^2-d+1 \ne 0$.

Therefore, the representation $\varphi '$ is irreducible. Consequently, it is a representation of Burau type.

Since the representation $\varphi '$ is of Burau type, thus, by Theorem 3, it is equivalent to $\chi (y)\otimes \beta _3(z)$ for some $y, z \in \mathbb {C}^*$. This implies that $\varphi '$ and $\chi (y)\otimes \beta _3(z)$ have the same eigenvalues which are $(1, d)$ and $(y, -yz)$ respectively. Thus, these representations are equivalent for

$$\begin{aligned} (y, z)=\left( d, -\frac{1}{d}\right) \hbox { and }(y, z)=(1, -d). \end{aligned}$$

On the other hand, if $d^2-d+1=0$, then the subspace $<(0, 1)>$ is invariant. This implies that the representation is reduced to a one dimensional representation.

Case 2. $\alpha (1-d)\ne 1$.

Consider the subspace $S=<u, v>$ where

$$\begin{aligned} u=e_1+\frac{1}{-1+\alpha -\alpha d}e_2 \hbox { and } v=e_1+\frac{\alpha }{-1+\alpha -\alpha d}e_2+\frac{1}{-1+\alpha -\alpha d}e_3. \end{aligned}$$

By direct computations, we have

• $\varphi (\sigma _1)(u)=(-1+\alpha +d-\alpha d)u$,

• $\varphi (\sigma _1)(v)=-u+v$,

• $\varphi (\sigma _2)(u)=v$ and

• $\varphi (\sigma _2)(v)=(-1-\alpha -d+\alpha d)u+(\alpha +d-\alpha d)v$.

Thus, the proper subspace S is invariant. Therefore, the representation $\varphi$ is reducible.

By restricting $\varphi$ to $S$, we obtain the representation $\varphi '$ defined as follows:

$$\begin{aligned} \varphi '(\sigma _1) = \left( \begin{array}{cc} -1+\alpha +d-\alpha d &{} -1 \\ 0 &{} 1 \\ \end{array} \right) \,\hbox { and } \varphi '(\sigma _2) = \left( \begin{array}{cc} 0 &{} 1-\alpha -d+\alpha d \\ 1 &{} \alpha +d-\alpha d \\ \end{array} \right) . \end{aligned}$$

Let $t=-1+\alpha +d-\alpha d$. The representation $\varphi '$ is reducible if and only if the matrices $\varphi '(\sigma _1)$ and $\varphi '(\sigma _2)$ have a common eigenvector. Direct computations show that the representation $\varphi '$ is irreducible if and only if $t^3\ne \pm 1$.

Therefore, any representation of type 1 is reduced to a Burau type representation (See Theorem 2.12).

Also, the representation $\varphi '$ is of Burau type, thus, by Theorem 3, it is equivalent to $\chi (y)\otimes \beta _3(z)$ for some $y, z \in \mathbb {C}^*$. Using the same argument of case 1, these representations are equivalent for:

$$\begin{aligned} (y, z)=\left( -1+\alpha +d-\alpha d, \frac{1}{1-\alpha -d+\alpha d}\right) \hbox { and }(y, z)=(1, 1-\alpha -d+\alpha d). \end{aligned}$$

$\square$

Proposition 4.2

Let $\varphi : B_3 \rightarrow GL_3(\mathbb {C})$ be a local representation of type 2 of $B_3$, then $\varphi$ is equivalent to a complex specialization of the standard representation.

Proof

Let $\varphi : B_3 \rightarrow GL_3(\mathbb {C})$ be a local representation of type 2 of $B_3$. We scale the basis using the matrix

$$\begin{aligned} P=Diag\left( \frac{1}{c}, 1, \gamma \right) . \end{aligned}$$

Thus, we get

$$\begin{aligned} P^{-1}\varphi (\sigma _1)P= \left( \begin{array}{ccc} 0 &{} bc &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0&{} 1 \\ \end{array} \right) \quad and \quad P^{-1}\varphi (\sigma _2)P= \left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} bc \\ 0 &{} 1&{} 0\\ \end{array} \right) . \end{aligned}$$

By letting $u=bc$, we notice that the representation $\varphi$ is equivalent to a complex specialization of the standard representation. $\square$

Now, we state the following theorem:

Theorem 4.3

Let $\varphi : B_3 \rightarrow GL_3(\mathbb {C})$ be a local representation of type 2 of $B_3$, then $\varphi$ is irreducible if and only if $bc \ne 1$.

5 Representations of dimension 6 of $B_3$

In this section, we study the irreducibility of the tensor product of two irreducible local representations of type 2 of $B_3$.

Consider two irreducible local representations $\rho _1=\varphi (b_1, c_1, \gamma _1)$ and $\rho _2=\varphi (b_2, c_2, \gamma _2)$ of type 2 of the braid group $B_3$.

These representations are defined as:

$$\begin{aligned} \rho _1(\sigma _1) = \left( \begin{array}{cc|cc} 0 &{} b_1 &{} 0 \\ c_1 &{} 0 &{} 0 \\ \hline 0 &{} 0 &{} 1 \\ \end{array} \right) , \rho _1(\sigma _2) = \left( \begin{array}{c|cc} 1 &{} 0 &{} 0 \\ \hline 0 &{} 0 &{} \frac{b_1c_1}{\gamma _1} \\ 0 &{} \gamma _1 &{} 0 \\ \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} \rho _2(\sigma _1) = \left( \begin{array}{cc|cc} 0 &{} b_2 &{} 0 \\ c_2 &{} 0 &{} 0 \\ \hline 0 &{} 0 &{} 1 \\ \end{array} \right) , \rho _2(\sigma _2) = \left( \begin{array}{c|cc} 1 &{} 0 &{} 0 \\ \hline 0 &{} 0 &{} \frac{b_2c_2}{\gamma _2} \\ 0 &{} \gamma _2 &{} 0 \\ \end{array} \right) , \end{aligned}$$

where $b_1c_1\ne 0, b_2c_2\ne 0, \gamma _1\ne 0$ and $\gamma _2\ne 0$.

By Proposition 4.2, the representations $\rho _1$ and $\rho _2$ are equivalent to the standard representations $\tau _1$ and $\tau _2$ defined by:

$$\begin{aligned} \tau _1(\sigma _1)= \left( \begin{array}{ccc} 0 &{} u_1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0&{} 1 \\ \end{array} \right) , \tau _1(\sigma _2)= \left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} u_1 \\ 0 &{} 1&{} 0\\ \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} \tau _2(\sigma _1)= \left( \begin{array}{ccc} 0 &{} u_2 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0&{} 1 \\ \end{array} \right) , \tau _2(\sigma _2)= \left( \begin{array}{ccc} 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} u_2 \\ 0 &{} 1&{} 0\\ \end{array} \right) . \end{aligned}$$

where $u_1=b_1c_1$, $u_2=b_2c_2$, $u_1\ne 1$ and $u_2\ne 1$.

Definition 5.1

Consider the tensor product $\tau _1\otimes \tau _2$ defined by $(\tau _1\otimes \tau _2)(\sigma _i)=\tau _1(\sigma _i)\otimes \tau _2(\sigma _i), i=1, 2$. We get the following matrices:

$$\begin{aligned} (\tau _1\otimes \tau _2)(\sigma _1) = \left( \begin{array}{ccccccccc} 0&{}0&{}0&{}0&{} u_1u_2&{}0&{}0&{}0&{}0 \\ 0 &{} 0 &{} 0&{}u_1&{}0 &{} 0 &{} 0&{}0&{}0 \\ 0 &{} 0 &{} 0&{}0&{}0 &{}u_1 &{} 0&{}0&{}0 \\ 0 &{} u_2 &{} 0&{}0&{}0 &{} 0 &{} 0&{}0&{}0 \\ 1 &{} 0 &{} 0&{}0&{}0 &{} 0 &{} 0&{}0&{}0 \\ 0 &{} 0 &{} 1&{}0&{}0 &{} 0 &{} 0&{}0&{}0 \\ 0 &{} 0 &{} 0&{}0&{}0 &{} 0 &{} 0&{}u_2&{}0 \\ 0 &{} 0 &{} 0&{}0&{}0 &{} 0 &{}1&{}0&{}0 \\ 0 &{} 0 &{} 0&{}0&{}0 &{} 0 &{} 0&{}0&{} 1 \\ \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} (\tau _1\otimes \tau _2)(\sigma _2) = \left( \begin{array}{ccccccccc} 1&{}0&{}0&{}0&{} 0 &{} 0&{}0&{}0&{}0 \\ 0 &{} 0 &{}u_2 &{}0&{}0 &{} 0 &{} 0&{}0&{}0 \\ 0 &{} 1 &{}0 &{}0&{}0 &{}0 &{} 0&{}0&{}0 \\ 0 &{} 0 &{} 0&{}0&{}0 &{} 0 &{} u_1&{}0&{}0 \\ 0 &{} 0 &{} 0&{}0&{}0 &{} 0 &{} 0&{}0&{} u_1u_2 \\ 0 &{} 0 &{} 0&{}0&{}0 &{} 0 &{} 0&{}u_1&{}0 \\ 0 &{} 0 &{} 0&{}1&{}0 &{} 0 &{} 0&{}0&{}0 \\ 0 &{} 0 &{} 0&{}0&{}0 &{} u_2 &{}0&{}0&{}0 \\ 0 &{} 0 &{} 0&{}0&{}1 &{} 0 &{} 0&{}0&{}0 \\ \end{array} \right) \end{aligned}$$

For simplicity, we denote $(\tau _1\otimes \tau _2)$ by $\rho$.

We now show that the representation $\rho$ is reducible.

Proposition 5.2

The representation $\rho$ is reducible.

Proof

By choosing a different basis for $\mathbb {C}^9$, namely $\{e_1, e_5, e_9, e_2, e_3, e_4, e_6, e_7, e_8\}$, the representation $\rho$ is equivalent to the representation $\psi$ whose matrices are given by:

$$\begin{aligned} \psi (\sigma _1) = \left( \begin{array}{ccccccccc} 0&{}u_1u_2&{}0&{}0&{} 0 &{} 0&{}0&{}0&{}0 \\ 1 &{} 0 &{} 0&{}0&{}0 &{} 0 &{} 0&{}0&{}0 \\ 0 &{} 0 &{} 1&{}0&{}0 &{}0 &{} 0&{}0&{}0 \\ 0 &{} 0 &{} 0&{}0&{}0 &{} u_1&{} 0&{}0&{}0 \\ 0 &{} 0 &{} 0&{}0&{}0 &{} 0 &{} u_1&{}0&{}0 \\ 0 &{} 0 &{} 0&{}u_2&{}0 &{} 0 &{} 0&{}0&{}0 \\ 0 &{} 0 &{} 0&{}0&{}1 &{} 0 &{} 0&{}0&{}0 \\ 0 &{} 0 &{} 0&{}0&{}0 &{} 0 &{}0&{}0&{}u_2 \\ 0 &{} 0 &{} 0&{}0&{}0 &{} 0 &{} 0&{}1&{}0 \\ \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} \psi (\sigma _2) = \left( \begin{array}{ccccccccc} 1&{}0&{}0&{}0&{} 0 &{} 0&{}0&{}0&{}0 \\ 0 &{} 0 &{}u_1u_2&{}0&{}0 &{} 0 &{} 0&{}0&{}0 \\ 0 &{}1 &{}0 &{}0&{}0 &{}0 &{} 0&{}0&{}0 \\ 0 &{} 0 &{} 0&{}0&{}u_2 &{} 0 &{} 0&{}0&{}0 \\ 0 &{} 0 &{} 0&{}1&{}0 &{} 0&{} 0&{}0&{} 0 \\ 0 &{} 0 &{} 0&{}0&{}0 &{} 0 &{} 0&{}u_1&{}0 \\ 0 &{} 0 &{} 0&{} 0 &{}0 &{} 0 &{} 0&{}0&{} u_1 \\ 0 &{} 0 &{} 0&{}0&{}0 &{} 1&{}0&{}0&{}0 \\ 0 &{} 0 &{} 0&{}0&{}0&{} 0 &{} u_2&{}0&{}0 \\ \end{array} \right) . \end{aligned}$$

It is clear from the form of the martices of the generators $\sigma _1$ and $\sigma _2$ that the representation $\psi$ is a direct sum of a standard representation and a representation $\varphi$ of dimension 6. $\square$

Definition 5.3

We define the representation $\varphi : B_3 \rightarrow GL_6(\mathbb {C})$ of $B_3$ of dimension 6 by:

$$\begin{aligned} \varphi (\sigma _1) = \left( \begin{array}{cccccc} 0&{}0 &{} u_1&{} 0&{}0&{}0 \\ 0&{}0 &{} 0 &{} u_1&{}0&{}0 \\ u_2&{}0 &{} 0 &{} 0&{}0&{}0 \\ 0&{}1 &{} 0 &{} 0&{}0&{}0 \\ 0&{}0 &{} 0 &{}0&{}0&{}u_2 \\ 0&{}0 &{} 0 &{} 0&{}1&{}0 \\ \end{array} \right) \end{aligned}$$

and

$$\begin{aligned} \varphi (\sigma _2) = \left( \begin{array}{cccccc} 0&{}u_2 &{} 0 &{} 0&{}0&{}0 \\ 1&{}0 &{} 0&{} 0&{}0&{} 0 \\ 0&{}0 &{} 0 &{} 0&{}u_1&{}0 \\ 0&{}0 &{} 0 &{} 0&{}0&{} u_1 \\ 0&{}0 &{} 1&{}0&{}0&{}0 \\ 0&{}0&{} 0 &{} u_2&{}0&{}0 \\ \end{array} \right) . \end{aligned}$$

We diagonalize the matrix corresponding to $\varphi (\sigma _1)$ by an invertible matrix, say T, and conjugate the matrix $\varphi (\sigma _2)$ by the same matrix T.

The invertible matrix T is given by

$$\begin{aligned} T= \left( \begin{array}{cccccc} 0&{}0 &{} 0 &{} 0&{}-\frac{\sqrt{u_1}}{\sqrt{u_2}}&{}\frac{\sqrt{u_1}}{\sqrt{u_2}} \\ -\sqrt{u_1}&{} \sqrt{u_1}&{} 0&{} 0&{}0&{} 0 \\ 0&{}0 &{} 0 &{} 0&{}1&{}1 \\ 1&{}1 &{} 0 &{} 0&{}0&{} 0 \\ 0&{}0 &{} -\sqrt{u_2}&{}\sqrt{u_2}&{}0&{}0 \\ 0&{}0&{} 1 &{} 1&{}0&{}0 \\ \end{array} \right) . \end{aligned}$$

In fact, computations show that

$$\begin{aligned} T^{-1}\sigma _1T= \left( \begin{array}{cccccc} -\sqrt{u_1}&{}0&{}0 &{} 0 &{} 0&{}0 \\ 0&{} \sqrt{u_1}&{} 0&{} 0&{}0&{} 0 \\ 0&{}0 &{} -\sqrt{u_2}&{}0&{}0&{}0 \\ 0&{}0 &{} 0 &{}\sqrt{u_2} &{}0&{} 0 \\ 0&{}0 &{} 0&{}0&{}-\sqrt{u_1}\sqrt{u_2}&{}0 \\ 0&{}0&{} 0 &{} 0&{}0&{}\sqrt{u_1}\sqrt{u_2} \\ \end{array} \right) . \end{aligned}$$

After conjugation, we get

$$\begin{aligned} T^{-1}\sigma _2T= \left( \begin{array}{cccccc} 0&{}0&{}\frac{u_1}{2} &{} \frac{u_1}{2} &{}\frac{1}{2\sqrt{u_2}}&{}-\frac{1}{2\sqrt{u_2}} \\ 0&{}0&{}\frac{u_1}{2} &{} \frac{u_1}{2} &{}-\frac{1}{2\sqrt{u_2}}&{}\frac{1}{2\sqrt{u_2}} \\ \frac{u_2}{2}&{}\frac{u_2}{2}&{}0 &{} 0 &{}-\frac{1}{2\sqrt{u_2}}&{}-\frac{1}{2\sqrt{u_2}} \\ \frac{u_2}{2}&{}\frac{u_2}{2}&{}0 &{} 0 &{}\frac{1}{2\sqrt{u_2}}&{}\frac{1}{2\sqrt{u_2}}\\ \frac{u_2\sqrt{u_2}}{2}&{}-\frac{u_2\sqrt{u_2}}{2}&{}-\frac{u_1\sqrt{u_2}}{2} &{} \frac{u_1\sqrt{u_2}}{2} &{}0&{}0 \\ -\frac{u_2\sqrt{u_2}}{2}&{}\frac{u_2\sqrt{u_2}}{2}&{}-\frac{u_1\sqrt{u_2}}{2} &{} \frac{u_1\sqrt{u_2}}{2} &{}0&{}0 \\ \end{array} \right) . \end{aligned}$$

For simplicity, we denote $T^{-1}\sigma _1T$ by $\sigma _1$ and $T^{-1}\sigma _2T$ by $\sigma _2$.

Assume that $u_i$, $i=1,2$, are non-zero complex numbers on the unit circle. We determine a sufficient condition for the irreducibility of the representation $\varphi$ of $B_3$ of dimension 6.

Lemma 5.4

Let $u_i$, $i=1,2$, be non-zero complex numbers on the unit circle. The representation $\varphi :B_3 \rightarrow GL_6(\mathbb {C})$ is irreducible if $\sqrt{u_1} \ne \pm \sqrt{u_2}$.

Proof

Direct computations show that $\sigma _i \sigma _i^*=I_6$, where $i=1,2$, * denotes the complex conjugate transpose, and $I_6$ denotes the $6 \times 6$ identity matrix.

Therefore, the representation is unitary. Consequently, if S is an invariant subspace then the orthogonal complement of S is also invariant.

Thus, it is sufficient to prove that there is no possible proper invariant subspace of dimensions 1,2,3.

Since the representations $\rho _1$ and $\rho _2$ are irreducible, then $u_1\ne 1$ and $u_2\ne 1$ (Lemmas 1 and 2). This implies that $\sqrt{u_1}\sqrt{u_2}\ne \pm \sqrt{u_1}$ and $\sqrt{u_1}\sqrt{u_2}\ne \pm \sqrt{u_2}$.

Let S be an invariant subspace of dimension $\le 3$.

We have the following cases:

Case 1. $S=<e_i>$, $i=1,...,6$.

Case 2. $S=<e_i, e_j>$, $i,j=1,...,6$, $i \ne j$.

Case 3. $S=<e_i, e_j, e_k>$, $i,j,k=1,...,6$, $i \ne j \ne k$.

In all the above cases, it is clear that $e_i \in S$ for some $1 \le i \le 6$.

As S is invariant, this implies that $\sigma _2(e_i) \in S$.

On the other hand, by direct computations, we have the following:

• $\sigma _2(e_1)= \frac{u_2}{2} (e_3+e_4) + \frac{u_2\sqrt{u_2}}{2}(e_5-e_6)$

• $\sigma _2(e_2)=\frac{u_2}{2} (e_3+e_4) + \frac{u_2\sqrt{u_2}}{2}(-e_5+e_6)$

• $\sigma _2(e_3)=\frac{u_1}{2} (e_1+e_2) - \frac{u_1\sqrt{u_2}}{2}(e_5+e_6)$

• $\sigma _2(e_4)=\frac{u_1}{2} (e_1+e_2) + \frac{u_1\sqrt{u_2}}{2}(e_5+e_6)$

• $\sigma _2(e_5)=\frac{1}{2\sqrt{u_2}} (e_1-e_2-e_3+e_4)$

• $\sigma _2(e_6)=\frac{1}{2\sqrt{u_2}} (-e_1+e_2 -e_3+e_4)$

In all the above cases, $\sigma _2(e_i) \notin S$ for all $1 \le i \le 6$. This gives a contradiction.

Therefore, there is no possible invariant subspace of dimension $\le 3$.

Thus, the representation $\varphi : B_3 \rightarrow GL_6(\mathbb {C})$ is irreducible if $\sqrt{u_1} \ne \pm \sqrt{u_2}$. $\square$

Now, we determine a necessary condition for irreducibility of the representation $\varphi : B_3 \rightarrow GL_6(\mathbb {C})$.

Lemma 5.5

Let $u_i$, $i=1,2$, be non-zero complex numbers on the unit circle. If $\sqrt{u_1} = \pm \sqrt{u_2}$, then the representation $\varphi : B_3 \rightarrow GL_6(\mathbb {C})$ is reducible.

Proof

We consider the following cases:

Case 1: $\sqrt{u_1} = \sqrt{u_2}$.

Let $S=$ $<e_5,u,v>$ where $u=e_1-e_3$ and $v= e_2-e_4$.

Direct computations show that:

• $\sigma _1(e_5)=-u_1e_5$

• $\sigma _2(e_5)=\frac{1}{2\sqrt{u_1}}(u - v)$

• $\sigma _1(u)=-\sqrt{u_1}u$

• $\sigma _2(u)=-\frac{u_1}{2}u-\frac{u_1}{2}v+u_1\sqrt{u_1}e_5$

• $\sigma _1(v)=\sqrt{u_1}v$

• $\sigma _2(v)=-\frac{u_1}{2}u-\frac{u_1}{2}v-u_1\sqrt{u_1}e_5$

Thus, the subspace S is invariant.

Case 2: $\sqrt{u_1} = - \sqrt{u_2}$

Let $S=$ $<e_5,u,v>$ where $u=e_1+e_4$ and $v= e_2+e_3$.

Direct computations show that:

• $\sigma _1(e_5)=u_1e_5$

• $\sigma _2(e_5)=-\frac{1}{2\sqrt{u_1}}(u - v)$

• $\sigma _1(u)=-\sqrt{u_1}u$

• $\sigma _2(u)=\frac{u_1}{2}u+\frac{u_1}{2}v-u_1\sqrt{u_1}e_5$

• $\sigma _1(v)=\sqrt{u_1}v$

• $\sigma _2(v)=\frac{u_1}{2}u+\frac{u_1}{2}v+u_1\sqrt{u_1}e_5$

Thus, the subspace S is invariant.

Therefore, the representation $\varphi : B_3 \rightarrow GL_6(\mathbb {C})$ is reducible if $\sqrt{u_1} = \pm \sqrt{u_2}$. $\square$

We state now the theorem of irreducibility of the considered representation.

Theorem 5.6

Let $u_i$, $i=1,2$, be non-zero complex numbers on the unit circle. The representation $\varphi : B_3 \rightarrow GL_6(\mathbb {C})$ is irreducible if and only if $\sqrt{u_1} \ne \pm \sqrt{u_2}$.