# 4-Dimensional projective planes of Lenz type III by Salzmann H. By Salzmann H.

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Then, the operations R in the group of the Hamiltonian applied to these functions yield Γ(n) (R)ki |n, k) , R|n, i) = k Γ(n ) (R)lj |n , l) . R|n , j) = l Since the operators and their representations are unitary, (n , j|R† = (n , j|R−1 = Γ(n ) (R)∗lj (n , l| , l we have (n , j|R−1 R|n, i) = (n , j|n, i) Γ(n ) (R)∗kj Γ(n) (R)li (n , k|n, l) . = kl 6 If we now sum both sides of this equation over the elements of the group of the Hamiltonian, and invoke the Great Orthogonality Theorem, we obtain (n , j|n, i) = |G|(n , j|n, i) R Γ(n ) (R)∗kj Γ(n) (R)li (n , k|n, l) = kl R |G| δn,n δk,l δi,j dn = |G|δn,n δi,j (n , k|n, k) , where |G| is the order of the group of the Hamiltonian and dn is the dimension of the nth irreducible representation.

An is (A1 A2 · · · An )ij = (A1 )ik1 (A2 )k1 k2 · · · (An )kn−1 j . ,kn−1 The corresponding matrix element of the transpose of this product is [(A1 A2 · · · An )t ]ij = (A1 A2 · · · An )ji . ,kn−1 We conclude that (A1 A2 · · · An )t = Atn Atn−1 · · · At1 9 and, similarly, that (A1 A2 · · · An )† = A†n A†n−1 · · · A†1 Group Theory Problem Set 5 November 6, 2001 Note: Problems marked with an asterisk are for Rapid Feedback. 1. 2, we established the relation Bi Bi† = I. Using the definitions in that proof, show that this result implies that Bi† Bi = I as well.

Da db = This shows that this direct product representation is, in fact, irreducible. It has dimensionality da db and the order of the direct product is, of course, |Ga | × |Gb |. 5. If the ϕi are orthonormal, and if this property is required to be preserved by the group of the Hamiltonian (as it must, to conserve probability), then, in Dirac notation, we have (i, j) ≡ ϕi (x)∗ ϕj (x) dx = = (i|R† R|j) = δi,j . , when written in matrix notation, Γ(R)† Γ(R) = I . Thus, the matrix representation is unitary.