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By Yu B.J., Xu M.

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Ti )acpp δbd11 . . δbqq d d − δca11 δca22 . . δcapp (Ti )db11 . . δbqq − . . − δca11 δca22 . . δcapp δbd11 . . (Ti )bqq . 35) with a relative minus sign between lines flowing in opposite directions. The reader will recognize this as the Leibnitz rule. 60): (λ) C λ Ti Ti = Cλ λ 00111100 11001100 11001100 00001111 T 00111100 11001100 000 111 000 111 = λ 11 00 00 11 00 11 00 11 00 11 00 11 λ11 00 00 11 0 1 1 0 0 1 11 00 00 11 00 11 00 11 00 11 00 11 . 8, March 2, 2008 35 DIAGRAMMATIC NOTATION The invariance conditions take a particularly suggestive form in the diagrammatic notation.

53) Hence, any matrix polynomial f (M) takes the scalar value f (λ i) on the Pi subspace f (M)Pi = f (λi )Pi . 54) This, of course, is the reason why one wants to work with irreducible reps: they reduce matrices and “operators” to pure numbers. 6 SPECTRAL DECOMPOSITION Suppose there exist several linearly independent invariant [d×d] hermitian matrices M1 , M2 , . , and that we have used M 1 to decompose the d-dimensional vector space V˜ = Σ ⊕ Vi . Can M2 , M3 , . . be used to further decompose V i ?

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, . . d, σ = 1, 2, . . dλ : ⎞⎫ ⎛ (Cλ )11 . . (Cλ )d1 ⎪ ⎬ ⎟ ⎜ ..

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A Biordered Set Representation of Regular Semigroups by Yu B.J., Xu M.


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