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where χ = χ ( θ$_{1}$, · · · θ$_{N}$ ) is an orthogonal basis. The integral element dµ$_{G}$ is the symmetry group invariance Haar measure for a specific symmetry group. The invariance Haar measure is essential in order to accommodate the composite internal symmetry. The color confine- ment is manifested by maintaining the color singlet state. Fortunately, the singlet state is the simplest case and its basis is given by χ$_{singlet}$ = 1. The prime objective is to confine the quarks in a color singlet state regardless of the composite internal group symmetry. 

## A. Unitary symmetry group U ( N ) 

In the context of the unitary symmetry group transformation, the physical Hamiltonian is subject to the Hermitian condition 

The unitary symmetry group subjects to the following constraint 

<img src='content_image/367995.jpg'>

<img src='content_image/367996.jpg'>

where the matrix U is a complex one and the Hermitian matrix is defined by U † = U ∗ $^{T}$. The unitary matrix U has 2 N 2 real parameters where N 2 comes from the real part of the unitary matrix and N 2 comes from the imaginary part. The unitary condition imposes [ N + 2 N ( N − 1) / 2] constraints on 2 N 2 real parameters to give N$_{adj}$ = N 2 essential real parameters. Hence, the unitary symmetry has N$_{fun}$ = N and N$_{adj}$ = N 2 for the fundamental and adjoint charges, respectively. 

The Gaussian unitary ensemble multiplied by the group volume element is defined in terms of Hermitian matrices as follows 

<img src='content_image/367994.jpg'>

which is invariant under automorphism unitary transformation. The invariance Haar mea- sure is given by the volume element 

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where H (0) kj and H (1) kj are real and imaginary parts of H$_{kj}$ . The partition function Z ( H ) is a function of only the eigenvalues θ$_{i}$ where i runs over 1 , · · · , N . This means that the 

