Expression of a Tensor Commutation Matrix in Terms of the Generalized Gell-Mann Matrices

We have expressed the tensor commutation matrix n\otimes n as linear combination of the tensor products of the generalized Gell-Mann matrices. The tensor commutation matrices 3\otimes 2 and 2\otimes 3 have been expressed in terms of the classical Gell-Mann matrices and the Pauli matrices.


Introduction
When we had worked on RAOELINA ANDRIAMBOLOLONA idea on the using tensor product in Dirac equation [1], [2] we had met the unitary matrix This matrix is frequently found in quantum information theory [3], [4], [5] where one write, by using the Pauli matrices [3], [4], [5] U 2⊗2 = 1 2 with I 2 the 2 × 2 unit matrix. We call this matrix a tensor commutation matrix 2 ⊗ 2. The tensor commutation matrix 3 ⊗ 3 is expressed by using the Gell-Mann matrices under the following form [6] 1 3 We have to talk a bit about different types of matrices because in the generalization of the above formulas we will consider the commutation matrix as a matrix of fourth order tensor and in expressing the commutation matrices U 3⊗2 , U 2⊗3 , at the last section, a commutation matrix will be considered as matrix of second order tensor. M m×n (C) denotes the set of m × n matrices whose elements are complex numbers.
1 Tensor product of matrices

Matrices
If the elements of a matrix are considered as the components of a second order tensor, we adopt the habitual notation for a matrix, without parentheses inside, whereas if the elements of the matrix are, for instance, considered as the components of sixth order tensor, three times covariant and three times contravariant, then we represent the matrix of the following way, for example  The first indices i 1 and j 1 are the indices of the outside parenthesis which we call the first order parenthesis ; the second indices i 2 and j 2 are the indices of the next parentheses which we call the second order parentheses ; the third indices i 3 and j 3 are the indices of the most interior parentheses, of this example, which we call third order parentheses. So, for instance, M 321 121 = 5. If we delete the third order parenthesis, then the elements of the matrix M are considered as the components of a forth order tensor, twice contravariant and twice covariant. A matrix is a diagonal matrix if deleting the interior parentheses we have a habitual diagonal matrix. A matrix is a symmetric (resp.antisymmetric) matrix if deleting the interior parentheses we have a habitual symmetric (resp. antisymmetric) matrix. We identify one matrix to another matrix if after deleting the interior parentheses they are the same matrix.

Tensor product of matrices
is called the tensor product of the matrix A by the matrix B. [3] ) where, i 1 i 2 are row indices j 1 j 2 are column indices.

Generalized Gell-Mann matrices
Let us fix n ∈ N, n ≥ 2 for all continuation. The generalized Gell-Mann matrices or n × n-Gell-Mann matrices are the traceless hermitian n × n ma- j the Kronecker symbol [7]. However, for the demonstration of the Theorem 3.2 below, denote, for 1 ≤ i < j ≤ n, the C 2 n = n! 2!(n−2)! n × n-Gell-Mann matrices which are symmetric with all elements 0 except the i-th row j-th column and the j-th row i-th column which are equal to 1, by Λ (ij) ; the C 2 n = n! 2!(n−2)! n × n-Gell-Mann matrices which are antisymmetric with all elements are 0 except the i-th row j-th column which is equal −i and the j-th row i-th column which is equal to i , by Λ [ij] and by Λ (d) ,1 ≤ d ≤ n − 1, the following (n − 1) n × n-Gell-Mann matrices which are diagonal : . . ., For n = 2 we have the Pauli matrices.

Tensor commutation matrices
For p, q ∈ N, p ≥ 2, q ≥ 2, we call tensor commutation matrices p ⊗ q the permutation matrix U p⊗q ∈ M pq×pq (C) formed by 0 and 1, verifying the property Considering U p⊗q as a matrix of a second order tensor, we can construct it by using the following rule [6].
Rule 3.1. Let us start in putting 1 at first row and first column, after that let us pass into second column in going down at the rate of p rows and put 1 at this place, then pass into third column in going down at the rate of p rows and put 1,and so on until there is only for us p − 1 rows for going down (then we have obtained as number of 1 : q. Then pass into the next column which is the (q + 1)-th column, put 1 at the second row of this column and repeat the process until we have only p − 2 rows for going down (then we have obtained as number of 1 : 2q). After that pass into the next column which is the (2q + 2) -th column, put 1 at the third row of this column and repeat the process until we have only p − 3 rows for going down (then we have obtained as number of 1 : 3q). Continuing in this way we will have that the element at p × q-th row and p × q-th column is 1. The other elements are 0.

Theorem 3.2. We have
where, i 1 i 2 are row indices j 1 j 2 are column indices [3]. Consider at first, the C 2 n symmetric n × n-Gell-Mann matrices which can be written Then The C 2 n antisymmetric n × n-Gell-Mann matrices can be written Then Now, consider the diagonal n × n-Gell-Mann matrices. Let d ∈ N, 1 ≤ d ≤ n − 1, is a diagonal matrix, so all that we have to do is to calculate the elements on the diagonal where l 1 = k 1 and l 2 = k 2 . Then, Let us distinguish two cases. 1 st case : k 1 = 1 or k 2 = 1 case1 : We can condense these cases in one formula which yields the diagonal of the diagonal matrix For all the n × n-Gell-Mann matrices we have for all l 1 , l 2 , k 1 ,k 2 ∈ {1, 2, . . . , n}.
Hence, by using (3.1) and the theorem is proved.