1 Introduction

Let \(B_n\) be the braid group on n strings. Ed. Formanek classified all irreducible representations of \(B_n\) of dimension at most n [3]. In [5], I. Sysoeva extended this classification to representations of dimensions n. It was shown that all irreducible complex specialization of the representations of \(B_n\) of dimension \(n \ge 9\) are equivalent to the tensor product of a one-dimensional representation and a specialization of the standard representations. It was proved that for \(n\ge 7\) every irreducible complex representation of \(B_n\) of corank two is equivalent to a specialization of the standard representation [5]. In [4], Larsen and Rowell proved that there are no irreducible unitary representations of \(B_n\) with dimension \(n + 1\) for \(n\ge 16\). In [6], I. Sysoeva proved that there are no irreducible representations of dimension \(n + 1\) for \(n \ge 10\).

In [2], Egea and Galina made a step forward in the classification of irreducible representations of the braid group \(B_n\). They introduced a new family of finite-dimensional complex representations of \(B_n\), which contains the standard representation. They gave a sufficient condition for members of this family to be irreducible. Moreover, they provided explicitly a subfamily of one parameter, self-adjoint representations \((\phi _m, V_m), 1 \le m < n\). The question of irreducibility of this family was further studied.

In our work, we determine a sufficient condition for the irreducibility of the family of representations of the braid group constructed by Egea and Galina without requiring that the representations are self-adjoint. Then, we construct a multi-parameter family of representations whose irreducibility will be studied.

In Sect. 2, we present the results of the paper of Egea and Galina [2]. More precisely, we define the family of finite-dimensional representations of the braid group, which was constructed in [2]. Then, we present a theorem, given in [2], which gives a sufficient condition for members of this family to be irreducible. In the hypothesis of the theorem, the representation is assumed to be self-adjoint. A specific family of representations was computed in their work.

In Sect. 3, we show that the self-adjoint condition required for members of this family is not needed to show their irreducibility. Moreover, we give explicitly another subfamily \((\psi _m, V_m)\), \( 1 \le m < n \), which contains non-self-adjoint representations. For all \( 1 \le m < n, \) \((\psi _m, V_m)\) is a multi-parameter representation, where dim \( V_m =\left( {\begin{array}{c}n\\ m\end{array}}\right) \). Finally, we study the irreducibility of \((\psi _m, V_m) \). More precisely, we show that for \( n> 2\) and \(n\ne 2m\), \((\psi _m, V_m)\) is an irreducible representation of \(B_n\) for all \( 1 \le m < n\), and if \(n=2m\), then \((\psi _m,V_m)\) is the sum of two representations of \(B_n\).

2 Definitions and known results

In this section, we list results of the paper of Egea and Galina. We present the family of finite-dimensional representations of the braid group containing the standard representation constructed in their work [2].

Definition 2.1

[1]. The braid group on n strings \(B_n\) is the abstract group with generators \(\tau _1,\dots ,\tau _{n-1}\) satisfying the following conditions:

$$\begin{aligned}&\tau _i\tau _{i+1}\tau _i = \tau _{i+1}\tau _i\tau _{i+1},i = 1,2,...,n-2\\&\quad \tau _i\tau _j = \tau _j\tau _i, |i- j| \ge 2. \end{aligned}$$

Definition 2.2

[7]. Let V be a finite-dimensional inner product space over \({\mathbb {C}}\) with inner product \(\langle \,,\rangle \). Given a linear operator \(T \in {\mathcal {L}}(V)\), the adjoint of T is defined to be the operator \(T^*\in {\mathcal {L}}(V)\) for which

$$\begin{aligned} \langle Tv, w\rangle = \langle v, T^*w\rangle , \,\text { for all}\, {v,w} \in V. \end{aligned}$$

A self-adjoint operator is an operator that is equal to its own adjoint. That is, T is self-adjoint if \(T=T^*.\) If, in addition, an orthonormal basis has been chosen, then the operator T is self-adjoint if and only if the matrix describing T with respect to this basis is Hermitian.

Definition 2.3

[7]. A self-adjoint representation \(\pi \) of a group G is a linear representation on a complex Hilbert space V, such that \(\pi (g)\) is a self-adjoint operator for every \(g \in G\).

Now, we introduce the family of representations of \(B_n\) constructed by Egea and Galina in [2]. Then, we present the main theorem that finds a sufficient condition for the irreducibility. A construction of such a family that satisfies the hypothesis of the theorem was made.

The authors in [2] considered n non-negative integers \(z_1, z_2, \dots , z_n\) not necessarily different; a set X which is the set of all the possible n-tuples obtained by permuting the coordinates of the fixed n-tuple \((z_1, z_2, \dots , z_n)\). For example, if \(n=3\), then

$$\begin{aligned} X=\{(z_1,z_2,z_3),(z_1,z_3,z_2),(z_2,z_1,z_3),(z_2,z_3,z_1)(z_3,z_1,z_2),(z_3,z_2,z_1)\}. \end{aligned}$$

They considered a complex vector space V with orthonormal basis \(\beta = \{v_x : x \in X\}\). The dimension of V is the cardinality of X. Then, they defined a representation \(\phi : B_n \rightarrow Aut(V)\), such that

$$\begin{aligned} \phi ( \tau _k)(v_x) = q_{x_k,x_{k+1}} v_{ \sigma _{k(x)}}. \end{aligned}$$

Here, \(q_{x_k,x_{k+1}}\) is a non-zero complex number that only depends on the places k and \(k+1\) of \(x = (x_1, \dots , x_n)\), and

$$\begin{aligned} \sigma _k (x_1, . . . , x_n) = (x_1, . . . , x_{k-1} , x_{k+1} , x_k, x_{k+2}, . . . , x_n). \end{aligned}$$

With these notations, the authors in [2] obtained the following theorems.

Theorem 2.4

[2]. (\(\phi \), V) is a representation of the braid group \(B_n.\)

Theorem 2.5

[2]. If \(\phi ( \tau _k)\) is a self-adjoint operator for all k, and for any pair \(x, y \in \) X, there exists j, 1 \(\le j \le n-1\), such that \(\vert q_{x_j,x_{j+1}}\vert ^2 \ne \vert q_{y_j,y_{j+1}}\vert ^2\), then (\(\phi \),V) is an irreducible representation of the braid group \(B_n\).

Example 2.6

[2] Let \(z_1,\dots ,z_n \in \{0,1\}\), such that \(z_1=z_2=\dots =z_m=1\) and \(z_{m+1}=\dots =z_{n}=0\). If \(V_m\) is the vector space with basis \(\beta _m=\{v_x,x\in X\}\), then dim\(V_m= \left( {\begin{array}{c}n\\ m\end{array}}\right) \). For each \(x=(x_1,\dots ,x_n)\in X\), let

$$\begin{aligned} q_{x_k, x_{k+1}} = \left\{ \begin{array}{lr} 1\quad if &{}\qquad x_k=x_{k+1}\\ t \quad if &{}\qquad x_k \ne x_{k+1} \end{array} \right. \end{aligned}$$

where t is real number, \(t\ne 0,1,-1\). Let \(\phi _m :B_n \rightarrow Aut(V_m)\) be a representation given by

$$\begin{aligned} \phi _m( \tau _k)(v_x) = q_{x_k,x_{k+1}} v_{ \sigma _{k(x)}}. \end{aligned}$$

The representation they obtained is self-adjoint. Thus, they used Theorem 2.2 to get the following result.

Theorem 2.7

[2]. Let \(n > 2\). Then, \((\phi _m, V_m)\) is an irreducible representation of \(B_n\), for all \(1 \le m < n\), such that \(2m \ne n\).

If \(n = 2m\) then \((\phi _m, V_m)\) is sum of two irreducible representations of \(B_n\).

3 Construction and main theorems

In this section, we deal with the representation in Sect. 2 without requiring the condition “\(\phi (\tau _k)\) is self- adjoint”, as stated in the hypothesis of Theorem 2.2 and we still obtain that the representation is irreducible. We then construct a multi-parameter representation of high degree, and study its irreducibility.

Theorem 3.1

If for any pair \(x, y \in X\), there exists k, 1 \(\le k \le n-1\), such that \( q_{x_k,x_{k+1}}q_{x_{k+1},x_k} \ne q_{y_k,y_{k+1}} q_{y_{k+1},y_k}\), then (\(\phi \),V) is an irreducible representation of the braid group \(B_n.\)

Proof

Let W be a non-zero invariant subspace of V. Consider \(\beta =\{v_x ;x\in X\}\) a basis for V. We now follow the steps adopted in [2] to show that if one of the basis vectors \(v_x\) belongs to W, then \(v_y\) belongs to W for any \(v_y\in \beta \). Since elements in X are obtained by permutation of a fixed element, it follows that any two elements are permutations of each other. Thus, for any two elements \(x,y\in X\), there exists a permutation \(\sigma \) formed of a product of transpositions, such that \(\sigma (x)=y \). For \(v_x\in W\). We let \(\sigma = \sigma _{i_1}\)...\(\sigma _{i_l} \). Then, \(\tau = \tau _{i_1}\dots \tau _{i_l}\) satisfies \(\phi (\tau )(v_x)=\lambda v_y\) for some non-zero complex number \(\lambda \). Hence, W contains \(v_y\), and, therefore, W contains the basis \(\beta \). Next, we show that \(v_x\) belongs to W for some \(x\in X\).

We have

$$\begin{aligned} (\phi (\tau _k))^2(v_x)=\phi (\tau _{k}) (q_{x_k, x_{k+1}} v_{\sigma _k (x)}) =q_{x_k, x_{k+1}} q_{x_{k+1},x_k} v_{\sigma ^2_k(x)}=q_{x_k,x_{k+1}} q_{x_{k+1},x_k}v_x. \end{aligned}$$

Hence, the matrices \((\phi (\tau _k))^2\) are diagonal in the basis \(\beta \) for all k,  \(1\le k \le n-1\). We now consider any of these diagonal matrices. Without loss of generality, we consider \((\phi (\tau _1))^2\). Since the matrix \((\phi (\tau _1))^2\) is diagonal, then any invariant subspace of V, in particular W, contains either an eigenvector \(v_x\) for some \(x\in X\) or a linear combination of its eigenvectors corresponding to the same eigenvalue. If \(v_x\in W\), then we are done. Otherwise, we assume that W contains a linear combination \(a_1v_{x^1}+a_2v_{x^2}+\dots +a_rv_{x^r}\), where at least two of the coefficients are non-zeros. Here, \(x^1,\dots , x^r \in X\) and \(v_{x^1}, \dots ,v_{x^r}\in \beta \) are the eigenvectors of \((\phi (\tau _1))^2\) corresponding to the same eigenvalue. We now show that if any linear combination of such vectors belongs to W, then one of the basis elements belongs to W. In particular, we prove that if any such linear combination of r vectors, with r more than one, belongs to W, then we can obtain a non-zero linear combination of at most \(r-1\) vectors that belongs to W. We consider

$$\begin{aligned} a_1v_{x^1}+a_2v_{x^2}+\cdots + a_rv_{x^r} \in W, \end{aligned}$$
(1)

where \(a_1,\dots ,a_r\) are different from zero. By hypothesis, for each pair of vectors in the basis \(\beta \), say \(v_{x^1}\) and \(v_{x^2}\), there exists k, 1 \(\le k \le n-1\), such that \( q_{x^1_k,x^1_{k+1}}q_{x^1_{k+1},x^1_k} \ne q_{x^2_k,x^2_{k+1}} q_{x^2_{k+1},x^2_k}\). Since \(a_1v_{x^1}+a_2v_{x^2}+\cdots +a_rv_{x^r} \in W \), it follows that:

$$\begin{aligned}&(\phi (\tau _k))^2(a_1v_{x^1}+a_2v_{x^2}+\cdots + a_rv_{x^r})\nonumber \\&\quad =a_1q_{x^1_k, x^1_{k+1}} q_{x^1_{k+1},x^1_k}v_{x^1}+a_2q_{x^2_k, x^2_{k+1}} q_{x^2_{k+1},x^2_k}v_{x^2}+\cdots + a_r q_{x^r_k ,x^r_{k+1}} q_{x^r_{k+1},x^r_k}v_{x^r}\in W. \end{aligned}$$
(2)

Multiplying (2) by \((q_{x^1_k,x^1_{k+1}}q_{x^1_{k+1},x^1_k})^{-1}\) and subtracting it from (1), we get

$$\begin{aligned}&(1-(q_{x^1_k , x^1_{k+1}} q_{x^1_{k+1},x^1_k})^{-1}(q_{x^2_k, x^2_{k+1}} q_{x^2_{k+1},x^2_k}))a_2 v_{x^2}+\cdots \\&\quad +(1-(q_{x^1_k,x^1_{k+1}} q_{x^1_{k+1},x^1_k})^{-1}(q_{x^r_k,x^r_{k+1}}q_{x^r_{k+1},x^r_k}))a_r v_{x^r}\in W. \end{aligned}$$

Having that

$$\begin{aligned} 1-(q_{x^1_k, x^1_{k+1}} q_{x^1_{k+1},x^1_k})^{-1}(q_{x^2_k, x^2_{k+1}}q_{x^2_{k+1},x^2_k})\ne 0, \end{aligned}$$

we get a non-zero linear combination of at most \(r-1\) vectors that belongs to W. Repeating this process, we obtain one of the basis vectors in W. Therefore, \(W=V\). \(\square \)

Remark

If \(\phi (\tau _k)\) is self-adjoint, then the condition \( q_{x_k,x_{k+1}}q_{x_{k+1},x_k}\) \(\ne q_{y_k,y_{k+1}} q_{y_{k+1},y_k}\) is equivalent to \(\vert q_{x_j,x_{j+1}}\vert ^2 \ne \vert q_{y_j,y_{j+1}}\vert ^2\), which is stated in the hypothesis of Theorem 2.2.

The previous theorem motivates the construction of new irreducible representations of the braid group. In [2], the authors constructed a subfamily of one parameter, self-adjoint representations \((\phi _m,V_m)\). We instead construct a subfamily of multi-parameter representations \((\psi _m, V_m)\) which contains non-self-adjoint representations.

We consider n non-negative integers \(z_1,z_2,\dots ,z_n\), not necessarily different; a set X which is the set of all the possible n-tuples obtained by permuting the coordinates of the fixed n-tuple \((z_1,z_2,\dots ,z_n)\). Let \(z_1,\dots ,z_n\in \{0,1\}\), such that \(z_1=z_2=\cdots z_m=1\) and \(z_{m+1}=\cdots =z_n=0\). Then, the cardinality of X is equal to

$$\begin{aligned} \left( {\begin{array}{c}n\\ m\end{array}}\right) = \dfrac{(n)!}{m!(n-m)!}. \end{aligned}$$

If \(V_m\) is the vector space with basis \(\beta _m=\{v_x, x\in X\}\), then dim \(V_m= \left( {\begin{array}{c}n\\ m\end{array}}\right) .\) For each \(x=(x_1,\dots ,x_n)\in X\), we let

$$\begin{aligned} q_{x_k, x_{k+1}} = \left\{ \begin{array}{lr} r_k \quad if &{}\qquad x_k=x_{k+1}\\ p_k \quad if &{}\qquad x_k > x_{k+1}\\ q_k \quad if&{}\qquad x_k < x_{k+1}, \end{array} \right. \end{aligned}$$

where \(r_k,p_k,q_k\in {\mathbb {R}} -\{0\}\), \(r_k^2\ne p_k q_k\), and \(p_kq_k>0\) for any k.

Let \(\psi _m :B_n \rightarrow Aut(V_m)\) be given by

$$\begin{aligned} \psi _m( \tau _k)(v_x) = q_{x_k,x_{k+1}} v_{ \sigma _{k(x)}}. \end{aligned}$$

We Now, consider the lexicographic order in X. Let \(n=5\) and \(m=3\). Then, we have dim\(V_m\) =10. The ordered basis in that case is

$$\begin{aligned} \beta :=\{v_{(0,0,1,1,1)},v_{(0,1,0,1,1)},v_{(0,1,1,0,1)}, v_{(0,1,1,1,0)},v_{(1,0,0,1,1)},v_{(1,0,1,0,1)}, \\ v_{(1,0,1,1,0)},v_{(1,1,0,0,1)}, v_{(1,1,0,1,0)},v_{(1,1,1,0,0)}\}, \end{aligned}$$

and the matrices in this basis are as follows:

$$\begin{aligned} \psi _{3(\tau _{1})}= \begin{pmatrix} r_1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} p_1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} p_1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} p_1 &{} 0 &{} 0 &{} 0 \\ 0 &{} q_1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} q_1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} q_1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} r_1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} r_1 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} r_1 \\ \end{pmatrix}, \end{aligned}$$
$$\begin{aligned} \psi _{3(\tau _{2})}= \begin{pmatrix} 0 &{} p_2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ q_2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} r_2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} r_2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} r_2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} p_2 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{}p_2 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} q_2 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} q_2 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} r_2 \\ \end{pmatrix}, \end{aligned}$$
$$\begin{aligned} \psi _{3(\tau _{3})}= \begin{pmatrix} r_3 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} p_3 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} q_3 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} r_3 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} p_3 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} q_3 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} r_3 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} r_3 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} p_3 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} q_3 &{} 0 \\ \end{pmatrix}, \end{aligned}$$
$$\begin{aligned} \psi _{3(\tau _{4})}= \begin{pmatrix} r_4 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} r_4 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} p_4 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} q_4 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} r_4 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} p_4 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} q_4 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} p_4 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} q_4 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} r_4 \\ \end{pmatrix}. \end{aligned}$$

As \(V_m\) is a complex vector space of finite dimension with an orthonormal basis \(\beta _m\), then using Definition 2.2, we get that \(\psi _m (\tau _k) \) is a self-adjoint operator iff \(q_{k}=p_k\) for any k, \(1\le k \le n-1 \) .

We now study the irreducibility of \((\psi _m, V_m)\). In what follows, \(p_k\) and \(q_k\) satisfy the conditions stated for \((\psi _m, V_m)\). That is, \(p_k,q_k\in {\mathbb {R}} -\{0\}\) and \(p_kq_k>0\) for any \(1\le k\le n-1\). To do so, we introduce some definitions and prove a lemma to prove Theorem 3.3.

Definition 3.2

Let \(x=(x_1,\dots ,x_n)\) and \(w=(w_1,\dots ,w_n)\) be any two elements in X. Let \(\sigma = \sigma _{i_1}\)...\(\sigma _{i_l} \) be a permutation sending x to w, where for \(1\le j \le l \) and \(1\le {i_j} < n\), \(\sigma _{i_j}\) is the transposition acting on an element \(x= (x_1,...,x_n),\) by swapping the entries \(x_{i_j}\) and \(x_{i_j+1}\), with \(x_{i_j}\ne x_{i_j+1}\). We introduce \(P_{x,w}\) a positive real number given by

$$\begin{aligned}&P_{x,w}=P (\sigma _{i_1})\dots P(\sigma _{i_l}),\mathrm {where} \\&\quad P(\sigma _{i_j})= \left\{ \begin{array}{lr} q_{i_j}p_{i_j}^{-1} \quad if \,\, \sigma _{i_j} \text { is acting on an element,} x=(x_1,...,x_n) \text { where} \\ x_{i_j}= 0 \text { and }x_{i_j+1}=1 \\ \\ p_{i_j}q_{i_j}^{-1} \quad if \,\,\, \sigma _{i_j} \text {is acting on an element} x=(x_1,...,x_n) \text { where} \\ x_{i_j}= 1 \text { and } x_{i_j+1}=0. \end{array} \right. \end{aligned}$$

Given the vectors x and w. If we have two permutations that send x to w, then the additional transpositions in the longer permutation will be even in number, and their corresponding values \(P(\sigma _{i_j})\)’s will be cancelled out. Thus, it is easy to see that \(P_{x,w}\) is independent of the choice of the permutation that sends x to w. For example, let \(x=(0,0,1,1)\) and \(w=(0,1,1,0)\). Consider the permutation \(\sigma = \sigma _3\sigma _2\) that sends x to w. Thus, we have

$$\begin{aligned} x=(0,0,1,1)\xrightarrow {\hbox {} \sigma _2 \hbox {}} (0,1,0,1)\xrightarrow {\text {} \sigma _3}(0,1,1,0)= w. \end{aligned}$$

We get that

$$\begin{aligned} P_{x,w}= P(\sigma _3)P(\sigma _2)=q_3p_3^{-1} q_2p_2^{-1}. \end{aligned}$$

Considering another permutation, \(\sigma = \sigma _1\sigma _3\sigma _1\sigma _2\) that also sends x to w. Then, we get

\(x=(0,0,1,1)\xrightarrow {\hbox { }\ \sigma _2} (0,1,0,1)\xrightarrow {\hbox { }\ \sigma _1}(1,0,0,1) \xrightarrow {\hbox { }\ \sigma _3} (1,0,1,0) \xrightarrow {\hbox { }\ \sigma _1} (0,1,1,0)= w.\)

Now, \(P_{x,w}=p_1q_1^{-1} q_3p_3^{-1} q_1p_1^{-1} q_2p_2^{-1}= q_3p_3^{-1} q_2q_2^{-1}.\)

We consider the case \(n=2m\) and we give the following definitions and lemmas.

Definition 3.3

For \(n=2m\), we observe that for any \(x\in X\), there is a \(y_x \in X \), where \(y_x\) is obtained from x by replacing the zeros by ones and the ones by zeros. For example

$$\begin{aligned} \text {if } x = (1, 0, 0, 1), \text {then}y_x = (0, 1, 1, 0). \end{aligned}$$

For any \(x \in X\), we define \(\lambda _x\) a non-zero positive real number defined as follows:

$$\begin{aligned} \lambda _{x}=\sqrt{P_{y_x,x}}. \end{aligned}$$

Equivalently, we define \(\lambda _{y_x}\) a positive real number, as

$$\begin{aligned} \lambda _{y_x}=\sqrt{P_{x,y_{x}}}. \end{aligned}$$

For example, we use the above definition to find \(\lambda _{(1,\dots ,1,0,\dots ,0)},\) where

\({(1,\dots ,1,0,\dots ,0)}\) is the last element in the ordered set X.

Example 3.4

Let X be the set of all the possible n-tuples obtained by permuting the coordinates of the fixed n-tuple \((z_1,z_2,\dots ,z_n)\), and \(z_1,\dots ,z_n\in \{0,1\}\), such that \(z_1=z_2= \cdots =z_m=1\) and \(z_{m+1}= \cdots =z_n=0\). For \(n=2m\),

let \(x=(0,\dots ,0,1,\dots ,1)\) and \(y_x=(1,\dots ,1,0,\dots ,0)\), be the first and last elements in the ordered set X, respectively. We consider a permutation that sends x to \(y_x\) in the following way: move each zero in the \((m-i)\)th place to the \((n-i)\)th place, starting from \(i=0\) till \(i=m-1\). This is done by a sequence of transpositions starting from \(\sigma _{m-i}\) till \(\sigma _{n-i-1}\), for each i. We then observe that this permutation will contain each of the transpositions \(\sigma _k\), k times for \(1\le k \le m\), and \(n -k\) times for \(m+1\le k \le n-1\). Having

$$\begin{aligned} P_{x,w}=P (\sigma _{i_1})\dots P(\sigma _{i_l}), \end{aligned}$$

we obtain that

$$\begin{aligned} \lambda _{(1,\dots ,1,0,\dots ,0)}=\sqrt{(q_1p_1^{-1})(q_2p_2^{-1})^2\dots (q_mp_m^{-1})^m (q_{m+1}p_{m+1}^{-1})^{m-1}\dots (q_{n-1}p_{n-1}^{-1})}. \end{aligned}$$

Lemma 3.5

For any two elements x and w in X, we have

$$\begin{aligned} \lambda _x=\lambda _{y_x}^{-1} \end{aligned}$$
(3)

and

$$\begin{aligned} \lambda _{x} P_{x, w}=\lambda _{w}. \end{aligned}$$
(4)

Proof

For any two elements x and w in X, let \(\sigma = \sigma _{i_1}\)...\(\sigma _{i_l} \) be the permutation that sends the element x to w. Since \(y_x\) is obtained from x by replacing the zeros with ones and the ones with zeros, and \(y_w\) is obtained from w in a similar way, it follows that \(\sigma \) also sends \(y_x\) to \(y_w\). Moreover, according to the definition of \(P_{x,w}\) and \(P_{y_x,y_w}\), we observe that \(P_{x,w}=P_{y_x,y_w}^{-1}\). Also, if \(\sigma \) is a permutation that sends the element x to w, then its inverse, \(\sigma ^{-1}=(\sigma _{i_1}\)...\(\sigma _{i_l})^{-1}= \sigma _{i_l}\)...\(\sigma _{i_1}\), is a permutation that sends w to x. The permutation \(\sigma ^{-1}\) will contain the same transpositions as that of \(\sigma \) but by reversing the order of zeros and ones for each transposition. In other words, if, in \(\sigma \), \(\sigma _{i_j}\) acts on an element where the \(i_j^{th}\) entry is 0 and \(i_{j+1}^{th}\) entry is 1, then in \(\sigma ^{-1}\), it will act on an element where the \(i_j^{th}\) entry is 1 and \(i_{j+1}^{th}\) entry is 0. Thus, we also have that \(P_{x,w}=P_{w,x}^{-1}\). In particular, \(P_{y_x,x}= P_{x,y_x}^{-1}\). Therefore, \(\lambda _x=\lambda _{y_x}^{-1} \). Given any three vectors x, y and w in X. Since \(P_{x,w}\) is independent of the choice of the permutation sending x to w, it follows that \(P_{y,w} P_{x,y}= P_{x,w}.\) We get that

$$\begin{aligned} \lambda _{x} P_{x, w}= & {} \sqrt{P_{{y_x},x}} P_{x,w}= \sqrt{P_{y_x,x}} \sqrt{P_{x,w}^2}= \sqrt{P_{y_x,x} P_{x,w} P_{x,w}}=\\= & {} \sqrt{P_{y_x,w} P_{x,w}}= \sqrt{P_{y_x,w} P_{y_x,y_w}^{-1}}=\sqrt{P_{y_x,w} P_{y_w,y_x}}=\sqrt{P_{y_w,w}}=\lambda _w. \end{aligned}$$

\(\square \)

Theorem 3.6

Let \( n> 2\). Then, \((\psi _m,V_m)\) is an irreducible representation of \(B_n\) for all \( 1 \le m < n\), such that \(n\ne 2m.\)

If \(n=2m\), then \((\psi _m,V_m)\) is the sum of two representations of \(B_n\).

Proof

Suppose that \(n \ne 2m\). In this case, we follow the steps used in [2] to show the irreducibility of \((\psi _m,V_m)\). Let \(x\ne y \in X\), then there exists i, \(1\le i\le n\), such that \(x_i\ne y_i\). If \(i >1\), we may suppose that \(x_{i-1}=y_{i-1}\); thus, \(q_{x_{i-1}, x_i}\ne q_{y_{i-1}, y_{i}}\) where one of them is equal to \(r_{i-1}\) and the other is either \(p_{i-1}\) or \(q_{i-1}\). Thus, \( q_{x_{i-1},x_{i}}q_{x_{i},x_{i-1}} \ne q_{y_{i-1},y_{i}} q_{y_{i},y_{i-1}},\) where one of them is equal to \(r_{i-1}^2\) and the other is equal to \(p_{i-1} q_{i-1}\).

If \(i=1\), \(x_1\ne y_1\), and \(n\ne 2m\), then there exists \(1\le l < n\), such that \(x_l\ne y_l\) and \(x_{l+1} = y_{l+1}\). Then, \(q_{x_l, x_{l+1}}\ne q_{y_l, y_{l+1}}\), where one of them is equal to \(r_l\) and the other is equal to \(p_{i-1}\) or \(q_{i-1}\). Thus, \( q_{x_l,x_{l+1}}q_{x_{l+1},x_l} \ne q_{y_l,y_{l+1}} q_{y_{l+1},y_l},\) where one of them is equal to \(r_l^2\) and the other is equal to \(p_l q_l\). Therefore, by Theorem 3.1, \(\psi _m\) is an irreducible representation.

For \(n=2m\), \(X=\{x, y_x : x\in Y , Y\subset X\}\), with \(y_x\) obtained from x by replacing the zeros by ones and the ones by zeros. The set Y can be considered as the set containing the first m elements of the ordered set X. For example, if

$$\begin{aligned} X:=\{ (0,0,1,1) ,(0,1,0,1),(0,1,1,0) ,(1,0,0,1),(1,0,1,0),(1,1,0,0)\}, \end{aligned}$$

then

$$\begin{aligned} Y:=\{(0,0,1,1), (0,1,0,1), (0,1,1,0)\}. \end{aligned}$$

Given \(x\in X\). It is easy to see that the vectors x and \(y=y_x\) have the property that \( q_{x_k,x_{k+1}}q_{x_{k+1},x_k} = q_{y_k,y_{k+1}} q_{y_{k+1},y_k}\) for any k, 1 \(\le k \le n-1\). Thus, the sufficient condition for irreducibility is not satisfied in the case \(n=2m\). Now, let \(\beta _1=\{ v_x+\lambda _{y_x}v_{y_x}; x\in Y \}\) and \(\beta _2=\{ v_x-\lambda _{y_x}v_{y_x}; x\in Y \}\). Let \(W_1\) and \(W_2\) be the vector spaces generated by \(\beta _1\) and \(\beta _2\), respectively. For \(x\in Y\), \(v_{x}+\lambda _{y_x}v_{y_x}\in W_1 \), then \(\lambda _{y_x}(\lambda _{y_x}^{-1}v_{x}+v_{y_x })\in W_1 \). Since \(\lambda _{x}=\lambda _{y_x}^{-1}\), it follows that \(v_{y_x}+\lambda _{x}v_{x} \in W_1 \). We obtain that \(v_{x}+\lambda _{y_x}v_{y_x}\in W_1\) for any \(x\in X.\) We claim that \(W_1\) and \(W_2\) are two invariant subspaces of \(V_m\). We prove that \(W_1\) is invariant and a similar proof follows for \(W_2\).

Let \(x=(x_1, x_2,\dots ,x_n)\in X\) and \(v_x+\lambda _{y_x}v_{y_x}\) be an element in \(W_1\). Given any \(1\le l < n\), and using Lemma 3.2, we consider the following cases:

Case 1: If \(x_l=x_{l+1} \), then \(y_{x_l}=y_{x_{l+1}}\). Thus

$$\begin{aligned} \phi (\tau _l)( v_x+\lambda _{y_x}v_{y_x})= r_l( v_x+\lambda _{y_x}v_{y_x})\in W_1. \end{aligned}$$

Case 2: If \(x_l=0\) and \(x_{l+1}=1 \), then \(y_{x_l}=1\) and \(y_{x_{l+1}}=0 \). Thus

$$\begin{aligned} \phi (\tau _l)( v_x+\lambda _{y_x}v_{y_x})= q_l v_{\sigma _l(x)}+ p_l\lambda _{y_x}v_{\sigma _l(y_x)}= q_l( v_{\sigma _l(x)}+p_lq_l^{-1}\lambda _{y_x}v_{\sigma _l(y_x)}). \end{aligned}$$

Since \(\sigma _l(y_x)=y_{\sigma _l(x)}\) and \( P_{y_x,\sigma _l(y_x)}=p_lq_l^{-1}\), it follows that:

$$\begin{aligned} p_lq_l^{-1}\lambda _{y_x}= \lambda _{y_x}\times P_{y_x,\sigma _l(y_x)}=\lambda _{\sigma _l(y_x)}. \end{aligned}$$

Therefore, \( q_l( v_{\sigma _l(x)}+p_lq_l^{-1}\lambda _{y_x}v_{\sigma _l(y_x)})\in W_1.\)

Case 3: If \(x_l=1\) and \(x_{l+1}=0 \), then \(y_{x_l}=0\) and \(y_{x_{l+1}}=1 \). Thus

$$\begin{aligned} \phi (\tau _l)( v_x+\lambda _{y_x}v_{y_x})= p_l v_{\sigma _l(x)}+ q_l\lambda _{y_x}v_{\sigma _l(y_x)}= p_l( v_{\sigma _l(x)}+q_lp_l^{-1}\lambda _{y_x}v_{\sigma _l(y_x)}). \end{aligned}$$

Since \( \sigma _l(y_x)=y_{\sigma _l(x)}\) and \( P_{y_x,\sigma _l(y_x)}=q_lp_l^{-1}\), it follows that:

$$\begin{aligned} q_lp_l^{-1} \lambda _{y_x}= \lambda _{y_x}\times P_{y_x,\sigma _l(y_x)} =\lambda _{\sigma _l(y_x)}. \end{aligned}$$

Therefore, \( p_l( v_{\sigma _l(x)}+q_lp_l^{-1}\lambda _{y_x}v_{\sigma _l(y_x)})\in W_1\).

Since \(\beta _1\) is a basis for \(W_1\), then the dimension of \(W_1\) is equal to cardinality of \(\beta _1,\) which is half that of \(V_m\). Thus, it is equal to \(\dfrac{\left( {\begin{array}{c}n\\ m\end{array}}\right) }{2}\) . \(\square \)

Example 3.7

We consider the representation (\(\psi _m, V_m)\) as previously defined, but for \(n=4\) and \(m=2\). We compute explicitly the two invariant subspaces \(W_1\) and \(W_2\) in this case. Consider the following set:

$$\begin{aligned} X= & {} \{(0,0,1,1),(0,1,0,1),(0,1,1,0),(1,0,0,1),(1,0,1,0),(1,1,0,0)\} \\ Y= & {} \{(0,0,1,1),(0,1,0,1),(0,1,1,0)\} \end{aligned}$$

and the ordered basis

$$\begin{aligned} \beta :=\{v_{(0,0,1,1)},v_{(0,1,0,1)},v_{(0,1,1,0)}, v_{(1,0,0,1)},v_{(1,0,1,0)}, v_{(1,1,0,0)}\}. \end{aligned}$$

The matrices in this basis are as follows:

$$\begin{aligned} \psi _{2(\tau _{1})}= \begin{pmatrix} r_1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} p_1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} p_1 &{} 0 \\ 0 &{} q_1 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} q_1 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} r_1 \\ \end{pmatrix}, \psi _{2(\tau _{2})}= \begin{pmatrix} 0 &{} p_2 &{} 0 &{} 0 &{} 0 &{} 0 \\ q_2 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} r_2 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} r_2 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} p_2 \\ 0 &{} 0 &{} 0 &{} 0 &{} q_2 &{} 0 \\ \end{pmatrix}, \end{aligned}$$
$$\begin{aligned} \psi _{2(\tau _{3})}= \begin{pmatrix} r_3 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} p_3 &{} 0 &{} 0 &{} 0 \\ 0 &{} q_3 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} p_3 &{} 0 \\ 0 &{} 0 &{} 0 &{} q_3 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} r_3 \\ \end{pmatrix}, \end{aligned}$$

where \(r_k,p_k,q_k\in {\mathbb {R}} -\{0\}\), \(r_k^2\ne p_k q_k\), \(p_k q_k>0\) for any k. Next, we compute \(\lambda _x\) for each x in X.

We consider the permutation \(P=\sigma _2\sigma _1\sigma _3\sigma _2\) that sends (0,0,1,1) to (1,1,0,0). Thus, for \(\lambda _{(1,1,0,0)}\) in \(\beta _1\), we have

$$\begin{aligned}&\lambda _{(1,1,0,0)}= \sqrt{(q_1p_1^{-1}) (q_2p_2^{-1})^2 (q_3p_3^{-1})}. \\&\quad \lambda _{(1,1,0,0)}= (q_1p_1^{-1})^{ \frac{1}{2}} (q_2p_2^{-1}) (q_3p_3^{-1})^{ \frac{1}{2}}. \end{aligned}$$

Note that \(\sigma _2 (1,1,0,0)=(1,0,1,0)\), then

$$\begin{aligned} \lambda _{(1,0,1,0)}= \lambda _{(1,1,0,0)}\times (p_2q_2^{-1})= (q_1p_1^{-1})^{ \frac{1}{2}}(q_3p_3^{-1})^{ \frac{1}{2}}. \end{aligned}$$

Since \(\sigma _3 (1,0,1,0)=(1,0,0,1)\), then

$$\begin{aligned} \lambda _{(1,0,0,1)}= \lambda _{(1,0,1,0)}\times (p_3q_3^{-1})= (q_1p_1^{-1})^{ \frac{1}{2}}(q_3p_3^{-1})^{ \frac{-1}{2}}. \end{aligned}$$

Thus, we have the following two invariant subspaces:

$$\begin{aligned}&W_1 =< v_{(0,0,1,1)}+(q_1p_1^{-1})^{ \frac{1}{2}} (q_2p_2^{-1}) (q_3p_3^{-1})^{\frac{1}{2}}v_{(1,1,0,0)},v_{(0,1,0,1)}+ \\&\quad (q_1p_1^{-1})^{ \frac{1}{2}}(q_3p_3^{-1})^{ \frac{1}{2}} v_{(1,0,1,0)}, v_{(0,1,1,0)}+ (q_1p_1^{-1})^{ \frac{1}{2}}(q_3p_3^{-1})^{ \frac{-1}{2}}v_{(1,0,0,1)}> \\&\quad W_2 =< v_{(0,0,1,1)}-(q_1p_1^{-1})^{ \frac{1}{2}} (q_2p_2^{-1}) (q_3p_3^{-1})^{\frac{1}{2}}v_{(1,1,0,0)},v_{(0,1,0,1)} \\&\quad -(q_1p_1^{-1})^{ \frac{1}{2}}(q_3p_3^{-1})^{ \frac{1}{2}} v_{(1,0,1,0)}, v_{(0,1,1,0)}- (q_1p_1^{-1})^{ \frac{1}{2}}(q_3p_3^{-1})^{ \frac{-1}{2}}v_{(1,0,0,1)}>. \end{aligned}$$

4 Conclusion

We weakened the conditions for the family of representations given by Egea and Galina to be irreducible. The sufficient condition obtained in our work is equivalent to that in [2] if we assume further that the representation is self-adjoint. We also constructed multi-parameter representations \((\psi _m,V_m)\) of the braid group \(B_n\), which satisfy the sufficient condition of irreducibility when \(n\ne 2m\) \((1\le m< n)\), and is the sum of two representations of \(B_n\) when \(n=2m\). Such irreducible representations can be useful in the progress of the classification of irreducible representations of \(B_n\).