Kronecker Permutation Matrix Essay

INTRODUCTION

This paper tries to solve a problem posed in [5], of searching a set K_3={τ_1,τ_2,⋯,τ_9 } of nine 3×3-matrices satisfying the following relation with the 3⨂3-Kronecker commutation matrix (KCM),
K_(3⨂3)=1/3 ∑_(i=1)^9▒τ_i ⊗τ_i (1)
The usefulness of the Kronecker permutation matrices, particularly the Kronecker commutation matrices (KCMs) in mathematical physics can be seen in [1], [2], [11]. In these papers, the 2⨂2- KCM is written in terms of the Pauli matrices, which are 2×2 matrices, by the following way
K_(2⊗2)= 1/2 ∑_(i=0)^3▒σ_i ⊗σ_i

The generalization of this formula in terms of generalized Gell-Mann matrices, which are a generalization of the Pauli matrices, is the topic of [6]. But there are other

is satisfied.

KRONECKER GENERALIZATION OF THE PAULI MATRICES

The Kronecker generalized Pauli matrices are the matrices (σ_i⨂σ_j )_(0≤i,j≤3)[10], [3], (σ_i⨂σ_j⨂σ_k )_(0≤i,j,k≤3), (σ_(i_1 )⨂σ_(i_2 )⨂⋯⨂σ_(i_n ) )_(0≤i_1,i_2,⋯,i_n≤3) [9] obtained by Kronecker product of the Pauli matrices and the 2×2 unit matrix. According to the propositions above, they are inverse-symmetric matrices and share many of the properties of the Pauli matrices: basis of C^(2^n×2^n ), ii., iii., iv., v. and vi. in the introduction.
Denote the set of (σ_(i_1 )⨂σ_(i_2 )⨂⋯⨂σ_(i_n ) )_(0≤i_1,i_2,⋯,i_n≤3) by K_(2^n ).

K_(2^n )={σ_0,σ_1,σ_2,σ_3 }^(⨂n)={σ_(i_1 )⨂σ_(i_2 )⨂⋯⨂σ_(i_n )/0≤i_1,i_2,⋯,i_n≤3}

The set
P_(2^n )=K_(2^n )⨂{1,-1,i,-i}

is a group called the Pauli group of (σ_(i_1 )⨂σ_(i_2 )⨂⋯⨂σ_(i_n ) )_(0≤i_1,i_2,⋯,i_n≤3) [9].
We can check easily, in using the relation σ_j σ_k=δ_jk σ_0+i∑_(i=1)^3▒ε_jkl σ_l, for j,k∈{1,2,3} that P_(2^n ) is equal to the set of the products of two elements of K_(2^n ), up to multiplicative phases, which are elements of