By D. B. Lichtenberg
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Example text
In other words , the tableau denotes a doublet. So far we have considered only four functions. But we have seen that there are 3! or six functions corresponding to tjJ(l23) and its permutations. The remaining two functions form the basis vectors of another doublet of S3 represented by the same tableau. In general, it can be shown that if an irreducible representation of Sn has N dimensions, there are N such representations of Sn. As another example, we consider the irreducible representations of S4.
Unless otherwise specified, we shall subsequently restrict ourselves to representations by matrices with nonvanishing determinants . The representations D(1), D(3), and D(4) are isomorphic to the group; D(2) is not. ]f a representation is isomorphic to the group it is said to be faithful. All groups of matrices are faithful representations of themselves. A representation is often called a vector represenfation to distinguish it from a ray representation in which, if a group element 9 i is represented by the matrix D(g i), then it is also represented by all the matrices e i• i D(gj), where (Xi is real.
Since all of the states of a multiplet are eigenstates of the Hamiltonian with the same energy eigenvalue, the states are said to be degenerate in the energy. The degeneracy is the number of states with a given energy. Jfthere is only one state corresponding to a given energy, the state is said to be singly degenerate, or more often, nondegenerate. As an example of a mUltiplet, we can consider the different states of a particle with sp in J. There exist 2J + 1 vectors representing different orientations of the spin of the particle with respect to an arb itrary z axis.