By Lucchhini A.

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57) j (i) Now the characteristic equation for M 2 (if nontrivial) can be used to decompose Vi subspace. 46) has more than one distinct eigenvalue; otherwise it is proportional to the unit matrix and commutes trivially with all group elements. A rep is said to be irreducible if all invariant matrices that can be constructed are proportional to the unit matrix. 40) is a statement that the defining rep is assumed irreducible. 31). 58) (remember that P i are also invariant [d×d] matrices). Hence, a [d×d] matrix rep can be written as a direct sum of [d i ×di ] matrix reps: G = 1G1 = Pi GPj = i,j Pi GPi = i Gi .

46) CMC = ⎜ ⎟. 0 0 ⎟ ⎜ .. . . ⎟ ⎜ . . ⎟ ⎜ ⎟ ⎜ 0 . . λ2 ⎟ ⎜ ⎜ λ3 . . ⎟ ⎠ ⎝ 0 0 .. . . 6). In the matrix C(M − λ2 1)C † the eigenvalues corresponding to λ 2 are replaced by zeroes: ⎞ ⎛ λ1 − λ2 ⎟ λ1 − λ2 ⎜ ⎟ ⎜ λ1 − λ2 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ . ⎟, ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ λ3 − λ2 ⎟ ⎜ ⎟ ⎜ λ3 − λ2 ⎠ ⎝ .. and so on, so the product over all factors (M − λ 2 1)(M − λ3 1) . . , with exception of the (M − λ1 1) factor, has nonzero entries only in the subspace associated with λ1 : ⎞ ⎛ 1 0 0 ⎟ ⎜ 0 1 0 0 ⎟ ⎜ ⎟ ⎜ 0 0 1 ⎟ ⎜ ⎟ ⎜ † 0 C (M − λj 1)C = (λ1 − λj ) ⎜ ⎟.

9) and the trace of a matrix ... 10) ... M can be drawn in the plane. Notation in which all internal lines are maximally crossed at each multiplication [318] is equally correct, but less pleasing to the eye. 2 CLEBSCH-GORDAN COEFFICIENTS Consider the product ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎞ 0 ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ C ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 0 1 1 1 0 0 0 .. 11) . 49). This matrix has nonzero entries only in the d λ rows of subspace Vλ . We collect them in a [dλ × d] rectangular matrix (C λ )α σ , α = 1, 2, .