# S3 Young Tableaux

1. Dec 13, 2013

### ChrisVer

Hello, I was "playing around" with the $S_{3}$ Young Tableaux, and trying to calculate the Young Operators for them... In the case of both antisymmetric and symmetric Young tableaux though, I had a problem...
for the standard YT I found:
1 2
3

I got $Y_{1}= e - (13) + (12) -(123)$

for
1 3
2

I got $Y_{2}=e + (13)-(12)-(132)$

(these are correct)

Then I tried the non standard YT:
3 2
1

for which I got: $Y_{3}=e+(23)-(13)-(132)$

My problem comes from the fact that $Y_{i} Y_{3}, (i=1,2)$ should give non zero value... In fact if someone tries
$Y_{2} Y_{3}$ he'll get ZERO
while for
$Y_{3} Y_{2}$ he'll get a nonzero value...
This is weird, because it means that:
$Y_{2}$ projecting the $Y_{3}\vec{v}$ will give zero (so $Y_{3}\vec{v}$ is orthogonal to $Y_{2}$'s subspace).
On the contrary:
$Y_{3}$ projecting the $Y_{2}\vec{w}$ will give something (so $Y_{2}\vec{w}$ is not orthogonal to $Y_{3}$'s subspace).
How can this be true?