Boson irreducible representations

The molecular spinors are expanded in terms of the four quaternion units (1, i, j, k). Two-dimensional contour maps of the large components are created in this work for the molecular spinors so to illustrate the nodal structure, and we review the relation between the quaternion representation and the normal four-component complex representation. [32] Each quaternion unit belongs to one of the boson irreducible representations (boson irreps) of C2v provided that the small components are neglected. 

The Pauli antisymmetry principle tells us that the wave function (including spin degrees of freedom), and thus the basis functions, for a system of identical particles must transform like the totally antisymmetric irreducible representation in the case of fermions, or spin (for odd k) particles, and like the totally symmetric irreducible representation in the case of bosons, or spin k particles (where k may take on only integer values). 

Equation (2.100) should be compared with Eq. (2.36). Several terms of Eq. (2.100) can be related by virtue of the fact that the Hamiltonian is a Hermitian operator, and by virtue of the eonservation of the total boson number, as stated by the invariance of the U(4) number operator, within a given irreducible representation. 

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