fundamental charges and N$_{adj}$ of adjoint charges. The unimodular constraint reduces N$_{fun}$ and N$_{adj}$ to N$_{fun}$ = N$_{fun}$ − 1 and N$_{adj}$ = N$_{adj}$ − 1. The N$_{adj}$ adjoint charges are invari- ance only over the N$_{fun}$ fundamental charges. The ( N$_{adj}$ − N$_{fun}$ ) extra real independent parameters do not appear in the Hamiltonian and they are integrated (washed) out. The unitary ensemble U ( N ) has N$_{fun}$ = N fundamental eigenvalues and N$_{adj}$ = N 2 parame- ters. The number of extra real independent parameters is N$_{adj}$ − N$_{fun}$ = ( N 2 − N ). This means that the unitary symmetry group has N fundamental chemical potentials and N 2 adjoint parameters. The N 2 adjoint parameters are transformed (diagonalized) to depend basically only on the N fundamental chemical potentials. The redundant ( N 2 − N ) pa- rameters are, subsequently, integrated out and they disappear from the resultant ensemble. The orthogonal ensemble has N$_{fun}$ = N fundamental eigenvalues and N$_{adj}$ = 1 $_{2}$N ( N + 1) adjoint parameters. The real symmetric N × N matrix has N$_{adj}$ − N$_{fun}$ = 1 $_{2}$N ( N − 1) redundant real parameters and these parameters are washed away by an appropriate trans- formation. The unitary symmetry group can be broken and decomposed to an orthogonal symmetry group with the same number of conserved charges N$_{fun}$ = N (or N$_{fun}$ = N − 1 when the unimodular constraint is embedded). The symmetry decomposition (breaking) from U ( N ) to orthogonal O$_{(}$$_{S}$$_{)}$ ( N ) leads to N ( N − 1) / 2 Goldstone bosons emerge as glue- balls (or gluon jets) in the medium. They escape from the Hagedorn bags and enrich the medium with gluonic contents and jets. The N ( N − 1) / 2 Goldstone bosons are identified as free colorless gluon degrees of freedom while the remaining 1 $_{2}$N ( N + 1) gluons remain as the exchange interacting gluons for the O$_{(}$$_{S}$$_{)}$ ( N ) symmetry group. The definitions of O$_{(}$$_{S}$$_{)}$ ( N ) and other symmetry groups such as the symplectic Sp ( N ) symmetry group shall be reviewed below in Sec. III. On the other hand, the symplectic ensemble has N$_{fun}$ = N eigenvalues of N × N quaternion-real matrix and N$_{adj}$ = N (2 N − 1). The number of re- dundant parameters in the symplectic symmetry group is 2 N ( N − 1). The (unimodular) unitary color U ( N$_{c}$ ) symmetry group may merge with other degree of freedom such as flavor symmetry U ( N$_{f}$ ) and the symmetry of the resultant composite transmutes to a symplectic (quaternion) symmetry group through the symmetry modification mechanism that is given by U ( N$_{c}$ ) × U ( N$_{f}$ ) → U ( N$_{c}$ + N$_{f}$ ) → O (2 N ) → Sp ( N ). Hence, under the preceding as- sumption, there is an indication that the eventual deconfined quark and gluon matter can be reached after a chain of multi-phase processes and this argument is not that simple. The thermodynamics of quark jets with an internal color structure has been considered 4