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 | | It's a curious way of defining elements in the group algebra of Sym(n) (over Q or C) that satissfy the commutation relations of su(m). |
 | | For example in the group algebra of Sym(3) define the three elements : h1,e12,e21 : 3*h1 = +2*(1,2)-(1,2,3)+(1,3,2)-2*(1,3) 3*e12 = +(2,3)-(1,2)+(1,2,3)-(1,3,2) 3*e21 = +(2,3)-(1,2,3)+(1,3,2)-(1,3) and the commutator [x,y]= x*y - y*x then you have : [h1,e21] = 2 e12, [h1,e12] = -2 e21, [e21,e12] = h1 which are the commutation relations of su(2)! |
 | | Then the image is a direct sum of matrix rings, so you can find elements of C[G] mapping to any set of elements in any of these matrix rings you'd like, such as the basis for sl(k,C) where k is the degree of the irreducible constituent in question. |
| www.math.niu.edu /~rusin/known-math/95/repres.sym (1234 words) |
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