- #1

- 78

- 0

## Main Question or Discussion Point

Greetings--I was wondering if someone could recommend some background reading on group theory in physics that explains group multiplication laws (I guess "tensor multiplication" is more appropriate).

I.e. in SO(3), 3 x 3 = 1 + 3 + 5

So that is to say if I had two vectors, [tex]\bar{x}=(x_1, x_2, x+3), \bar{y} = (y_1, y_2, y_3)[/tex], then the tensor product of the two vectors is equal to a direct sum of something that transforms as a singlet, triplet, and 5-let in SO(3). I guess the singlet ends up being the trace ([tex]\sum x_iy_i[/tex]), and the triplet ends up being the cross product, while the 5-let is some linear combination of the left over pairs. How do I construct these? (i.e. in components, what *are* the 5-let components?)

I.e. in SO(3), 3 x 3 = 1 + 3 + 5

So that is to say if I had two vectors, [tex]\bar{x}=(x_1, x_2, x+3), \bar{y} = (y_1, y_2, y_3)[/tex], then the tensor product of the two vectors is equal to a direct sum of something that transforms as a singlet, triplet, and 5-let in SO(3). I guess the singlet ends up being the trace ([tex]\sum x_iy_i[/tex]), and the triplet ends up being the cross product, while the 5-let is some linear combination of the left over pairs. How do I construct these? (i.e. in components, what *are* the 5-let components?)