okay so i was reading a book on representations and found this discussion and was confused:(adsbygoogle = window.adsbygoogle || []).push({});

http://books.google.com/books?id=Hm...eory and physics&pg=PA53#v=onepage&q=&f=false

it starts at the bottom of pg 53 and ends at the top of pg 54

so I understood the beginning of the discussion:

there are matrix representations of S3 and they permute the vector components

but (1,1,1) constitutes an invariant subspace cause what ever you permutation occurs on that, will bring you back to (1,1,1)

but then it goes on to say:

"To find another invariant subspace, we note that every 3 X 3 matric in the representation belongs to O(3) and hence preserves the ordinary Euclidean scalar product. Therefore, the subspace W' orthogonal to (1,1,1) is also invariant."

It then goes on to list the invariant subspace.

I got lost. Can anyone help? Why did they come up with those numbers? (and how too)

The example is particularly important cause he uses it later:

http://books.google.com/books?id=Hm...eory and physics&pg=PA96#v=onepage&q=&f=false

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# Irreducible representation of S3

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