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S3 Young Tableaux

  1. Dec 13, 2013 #1

    ChrisVer

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    Gold Member

    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...
    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?
     
  2. jcsd
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