Hello, I was "playing around" with the [itex]S_{3}[/itex] 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...(adsbygoogle = window.adsbygoogle || []).push({});

for the standard YT I found:

1 2

3

I got [itex]Y_{1}= e - (13) + (12) -(123)[/itex]

for

1 3

2

I got [itex]Y_{2}=e + (13)-(12)-(132)[/itex]

(these are correct)

Then I tried the non standard YT:

3 2

1

for which I got: [itex]Y_{3}=e+(23)-(13)-(132)[/itex]

My problem comes from the fact that [itex]Y_{i} Y_{3}, (i=1,2)[/itex] should give non zero value... In fact if someone tries

[itex]Y_{2} Y_{3}[/itex] he'll get ZERO

while for

[itex]Y_{3} Y_{2}[/itex] he'll get a nonzero value...

This is weird, because it means that:

[itex]Y_{2}[/itex] projecting the [itex]Y_{3}\vec{v}[/itex] will give zero (so [itex]Y_{3}\vec{v}[/itex] is orthogonal to [itex]Y_{2}[/itex]'s subspace).

On the contrary:

[itex]Y_{3}[/itex] projecting the [itex]Y_{2}\vec{w}[/itex] will give something (so [itex]Y_{2}\vec{w}[/itex] is not orthogonal to [itex]Y_{3}[/itex]'s subspace).

How can this be true?

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# S3 Young Tableaux

Can you offer guidance or do you also need help?

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