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The unimodular-like orthogonal O$_{(}$$_{S}$$_{)}$ ( N$_{c}$ ) symmetry group is constructed by imposing the unimodular-like constraint ∑ N$_{c}$ i $_{=1}$θ$_{i}$ = 0 to the standard orthogonal symmetry group. The color-singlet Gaussian integral for the unimodular-like orthogonal symmetry group S O$_{(}$$_{S}$$_{)}$ ( N ) is related to O$_{(}$$_{S}$$_{)}$ ( N ) in the following way 

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## C. Symplectic symmetry group Sp ( N ) (N is the number of quaternion) 

The symplectic symmetry group is time-reversal invariance. The anti-symmetrization in the symmetry group Sp ( N ) leads to odd-spin. The algebra of the symplectic symmetry group can be expressed in terms of quaternions. The quaternions are convenient to deal with the anti-symmetrization. The notation Sp ( N ) is equivalent to Sp (2 N, R ) where N corresponds to the number of quaternions. 

The transformation matrix U of the Sp ( N ) group is of the size of (2 N × 2 N ). This transformation matrix is considered as cut into N 2 blocks of 2 × 2 and each 2 × 2 block is expressed in terms of the quaternions. In this sense the number of real independent parameters these are defined in the ( 2 N × 2 N ) self-dual Hermitian matrix is given by the number of the adjoint variables N$_{adj}$ . The number of the fundamental (energy-) eigen variables is N$_{fun}$ = N while the number of the adjoint variables is N$_{adj}$ = N (2 N − 1). The invariance Haar measure for the symplectic symmetry group Sp ( N ) with the circular canonical ensemble is given by 

The normalisation constant N$_{Sp}$ is determined by 

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Under extreme conditions, the saddle points of the group continuous variables accumulate around the origin. Therefore, the canonical ensemble is approximated to the Gaussian-like 

