Re: irreducible representation: symmetric group S3

On Jul 6, 9:24 am, Mariano Suárez-Alvarez
<mariano.suarezalva...@xxxxxxxxx> wrote:
On Jul 6, 12:58 am, ganesh <ganeshs...@xxxxxxxxx> wrote:

hi,
   I was trying to understand the irreducible representation of the
symetric group S3.
In a three dimensional space, spanned by e1, e2 and e3 (say, the x,y
and z axis),
I can have a straight diagonal line passing through the origin (i.e
x=y=z). Now, whatever permutaion I apply to this space, the above
straight line remains invariant. So, this straight line becomes a one
dimensional invariant vector space. This vector space leads to a one
dimensional irreducible representation of S3.

    What about the other 1D and 2D irreducible representation for S3?
How do I picture it geometrically? Does the 2D irreducible
representation imply that there exists a place, which will remain
invariant to any permutaion of the axis (x, y, z)?

The orthogonal complement (with respect to the 'usual'
inner product) to the straight diagonal line you
found above is a two-dimensional irreducible representation.

The other representation is the sign representation...

-- m

understood the 2D thing.... thanks....
The sign representation would refer to the even and odd permutation
right? is there a geometric visualization of the sign representation?
.

Relevant Pages