On the representations of the braid group constructed by C. M. Egea and E. Galina

In this paper, we determine a sufficient condition for the irreducibility of the family of representations of the braid group constructed by C. M. Egea and E. Galina without requiring that the representations are self-adjoint. Then, we construct a new subfamily of multi-parameter representations (ψm,Vm),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\psi _m,V_m), $$\end{document}1≤m<n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le m< n$$\end{document}, of dimension Vm=nm\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ V_m =\left( {\begin{array}{c}n\\ m\end{array}}\right) $$\end{document}. Finally, we study the irreducibility of (ψm,Vm)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\psi _m,V_m) $$\end{document}.


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 ≥ 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 ≥ 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 ≥ 16. In [6], I. Sysoeva proved that there are no irreducible representations of dimension n + 1 for n ≥ 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 (φ m , V m ), 1 ≤ 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 (ψ m , V m ), 1 ≤ m < n, which contains non-self-adjoint representations. For all 1 ≤ m < n, (ψ m , V m ) is a multi-parameter representation, where dim V m = n m . Finally, we study the irreducibility of (ψ m , V m ). More precisely, we show that for n > 2 and n = 2m, (ψ m , V m ) is an irreducible representation of B n for all 1 ≤ m < n, and if n = 2m, then (ψ m , V m ) is the sum of two representations of B n .

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 τ 1 , . . . , τ n−1 satisfying the following conditions: Definition 2.2 [7]. Let V be a finite-dimensional inner product space over C with inner product , . Given a linear operator T ∈ L(V ), the adjoint of T is defined to be the operator T * ∈ L(V ) for which 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 π of a group G is a linear representation on a complex Hilbert space V , such that π(g) is a self-adjoint operator for every g ∈ 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 , . . . , 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 , . . . , z n ). For example, if n = 3, then They considered a complex vector space V with orthonormal basis β = {v x : x ∈ X }. The dimension of V is the cardinality of X . Then, they defined a representation φ : 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 , . . . , x n ), and With these notations, the authors in [2] obtained the following theorems.
Theorem 2.7 [2]. Let n > 2. Then, (φ m , V m ) is an irreducible representation of B n , for all 1 ≤ m < n, such that 2m = n. If n = 2m then (φ m , V m ) is sum of two irreducible representations of B n .

Construction and main theorems
In this section, we deal with the representation in Sect. 2 without requiring the condition "φ(τ 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.
x ∈ 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 ∈ β. 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 ∈ X , there exists a permutation σ formed of a product of transpositions, such that σ (x) = y.
For v x ∈ W . We let σ = σ i 1 ...σ i l . Then, τ = τ i 1 . . . τ i l satisfies φ(τ )(v x ) = λv y for some non-zero complex number λ. Hence, W contains v y , and, therefore, W contains the basis β. Next, we show that v x belongs to W for some x ∈ X . We have Hence, the matrices (φ(τ k )) 2 are diagonal in the basis β for all k, 1 ≤ k ≤ n − 1. We now consider any of these diagonal matrices. Without loss of generality, we consider (φ(τ 1 )) 2 . Since the matrix (φ(τ 1 )) 2 is diagonal, then any invariant subspace of V , in particular W , contains either an eigenvector v x for some x ∈ X or a linear combination of its eigenvectors corresponding to the same eigenvalue.
If v x ∈ W , then we are done. Otherwise, we assume that W contains a linear combination where at least two of the coefficients are non-zeros. Here, x 1 , . . . , x r ∈ X and v x 1 , . . . , v x r ∈ β are the eigenvectors of (φ(τ 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 where a 1 , . . . , a r are different from zero. By hypothesis, for each pair of vectors in the basis β, say v x 1 and Having that 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 .
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 (φ m , V m ). We instead construct a subfamily of multi-parameter representations (ψ m , V m ) which contains non-self-adjoint representations.
We now study the irreducibility of (ψ m , V m ). In what follows, p k and q k satisfy the conditions stated for (ψ m , V m ). That is, p k , q k ∈ R − {0} and p k q k > 0 for any 1 ≤ k ≤ n − 1. To do so, we introduce some definitions and prove a lemma to prove Theorem 3.3. x = (x 1 , . . . , x n ) and w = (w 1 , . . . , w n ) be any two elements in X. Let σ = σ i 1 ...σ i l be a permutation sending x to w, where for 1 ≤ j ≤ l and 1 ≤ i j < n, σ 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 = x i j +1 . We introduce P x,w a positive real number given by 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(σ 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 σ = σ 3 σ 2 that sends x to w. Thus, we have

Definition 3.2 Let
We get that Considering another permutation, σ = σ 1 σ 3 σ 1 σ 2 that also sends x to w. Then, we get x = (0, 0, 1, 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 ∈ X , there is a y x ∈ X , where y x is obtained from x by replacing the zeros by ones and the ones by zeros. For example if x = (1, 0, 0, 1), then y x = (0, 1, 1, 0).

Lemma 3.5 For any two elements x and w in X , we have
and Proof For any two elements x and w in X , let σ = σ i 1 ...σ 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 σ 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 −1 y x ,y w . Also, if σ is a permutation that sends the element x to w, then its inverse, σ −1 = (σ i 1 ...σ i l ) −1 = σ i l ...σ i 1 , is a permutation that sends w to x. The permutation σ −1 will contain the same transpositions as that of σ but by reversing the order of zeros and ones for each transposition. In other words, if, in σ , σ i j acts on an element where the i th j entry is 0 and i th j+1 entry is 1, then in σ −1 , it will act on an element where the i th j entry is 1 and i th j+1 entry is 0. Thus, we also have that P x,w = P −1 w,x . In particular, P y x ,x = P −1 x,y x . Therefore, λ x = λ −1 y x . 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 λ x P x,w = P y x ,x P x,w = P y x ,x P 2 x,w = P y x ,x P x,w P x,w = = P y x ,w P x,w = P y x ,w P −1 y x ,y w = P y x ,w P y w ,y x = P y w ,w = λ w .
Theorem 3.6 Let n > 2. Then, (ψ m , V m ) is an irreducible representation of B n for all 1 ≤ m < n, such that n = 2m. If n = 2m, then (ψ m , V m ) is the sum of two representations of B n .
Proof Suppose that n = 2m. In this case, we follow the steps used in [2] to show the irreducibility of (ψ m , V m ). Let x = y ∈ X , then there exists i, 1 ≤ i ≤ n, such that x i = y i . If i > 1, we may suppose that x i−1 = y i−1 ; thus, q x i−1 ,x i = 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 .
where one of them is equal to r 2 i−1 and the other is equal to p i−1 q i−1 . If i = 1, x 1 = y 1 , and n = 2m, then there exists 1 ≤ l < n, such that x l = y l and x l+1 = y l+1 . Then, q x l ,x l+1 = 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 = q y l ,y l+1 q y l+1 ,y l , where one of them is equal to r 2 l and the other is equal to p l q l . Therefore, by Theorem 3.1, ψ m is an irreducible representation.
For n = 2m, X = {x, y x : x ∈ Y, Y ⊂ 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 Given x ∈ 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 ≤ k ≤ n − 1. Thus, the sufficient condition for irreducibility is not satisfied in the case n = 2m. Now, let β 1 = {v x + λ y x v y x ; x ∈ Y } and β 2 = {v x − λ y x v y x ; x ∈ Y }. Let W 1 and W 2 be the vector spaces generated by β 1 and β 2 , respectively. For We obtain that v x + λ y x v y x ∈ W 1 for any x ∈ 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 , . . . , x n ) ∈ X and v x + λ y x v y x be an element in W 1 . Given any 1 ≤ l < n, and using Lemma 3.2, we consider the following cases: Case 2: If x l = 0 and x l+1 = 1, then y x l = 1 and y x l+1 = 0. Thus Since σ l (y x ) = y σ l (x) and P y x ,σ l (y x ) = p l q −1 l , it follows that: Therefore, q l (v σ l (x) + p l q −1 l λ y x v σ l (y x ) ) ∈ 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 Since σ l (y x ) = y σ l (x) and P y x ,σ l (y x ) = q l p −1 l , it follows that: Therefore, p l (v σ l (x) + q l p −1 l λ y x v σ l (y x ) ) ∈ W 1 . Since β 1 is a basis for W 1 , then the dimension of W 1 is equal to cardinality of β 1 , which is half that of V m .
Thus, it is equal to n m 2 .

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 (ψ m , V m ) of the braid group B n , which satisfy the sufficient condition of irreducibility when n = 2m (1 ≤ 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 .
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