Mathematical Methods for Physics and Engineering - Matematica.NET

REPRESENTATION THEORY◮For the group S 3 of permutations on three objects, which has group multiplication table28.8 on p. 1055, with (in cycle notation)I = (1)(2)(3), A =(123), B =(132C = (1)(2 3), D = (3)(1 2), E = (2)(1 3),use as the components of a basis vector the ordered letter tripletsu 1 = {PQR}, u 2 = {QRP}, u 3 = {RPQ},u 4 = {PRQ}, u 5 = {QPR}, u 6 = {RQP}.Generate a six-dimensional representation D = {D(X)} of the group and confirm that therepresentative matrices multiply according to table 28.8, e.g.D(C)D(B) =D(E).It is immediate that the identity permutation I = (1)(2)(3) leaves all u i unchanged, i.e.u ′ i = u i for all i. The representative matrix D(I) is thus I 6 ,the6× 6 unit matrix.We next take X as the permutation A = (1 2 3) and, using (29.1), let it act on each ofthe components of the basis vector:u ′ 1 = Au 1 =(123){PQR} = {QRP} = u 2u ′ 2 = Au 2 =(123){QRP} = {RPQ} = u 3..u ′ 6 = Au 6 =(123){RQP} = {QPR} = u 5 .The matrix M(A) has to be such that u ′ = M(A)u (here dots replace zeros to aid readability):⎛u ′ =⎜⎝u 2u 3u 1u 6u 4u 5⎞ ⎛=⎟ ⎜⎠ ⎝· 1 · · · ·· · 1 · · ·1 · · · · ·· · · · · 1· · · 1 · ·· · · · 1 ·⎞ ⎛⎟ ⎜⎠ ⎝u 1u 2u 3u 4u 5u 6⎞≡ M(A)u.⎟⎠D(A) isthenequaltoM T (A).The other D(X) are determined in a similar way. In general, ifXu i = u j ,then [M(X)] ij = 1, leading to [D(X)] ji =1and[D(X)] jk =0fork ≠ i. For example,Cu 3 = (1)(23){RPQ} = {RQP} = u 6implies that [D(C)] 63 =1and[D(C)] 6k =0fork =1, 2, 4, 5, 6. When calculated in full⎛⎞⎛⎞· · · 1 · ·· 1 · · · ·· · · · 1 ·· · 1 · · ·· · · · · 11 · · · · ·D(C) =⎜ 1 · · · · ·, D(B) =⎟⎜ · · · · · 1,⎟⎝ · 1 · · · · ⎠⎝ · · · 1 · · ⎠· · 1 · · ·· · · · 1 ·⎛⎞· · · · · 1· · · 1 · ·· · · · 1 ·D(E) =⎜ · 1 · · · ·,⎟⎝ · · 1 · · · ⎠1 · · · · ·1080