Reading off tensor index symmetries from a Young Tableau

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I'm reading a paper which has given the index symmetries in terms of a Young Tableau which I'm having trouble understanding e.g. one is of the form



$[mu][nu]$

$[rho][sigma]$



I understand that if it was fully horizontal (vertical) then it would correspond to a totally (anti)symmetric tensor in all its indices. In this example I'm given to believe it is actually ambiguous in that it could correspond to a tensor which is either symmetric in exchange of $munu$ and $rhosigma$ i.e. in the bracket notation then it'd be $T^munurhosigma$ = $T^(munu)(rhosigma)$, but also in fact could be used to represent a tensor which is antisymmetric in $murho$ and $nusigma$ i.e. $T^murhonusigma$ = $T^[murho][nusigma]$. Is this the case or have I misunderstood?



Also in a slightly more complicated case, say



$[mu][nu][lambda]$

$[rho][sigma]$



Can this be read as a tensor symmetric in $munulambda$ and $rho sigma$, or a tensor which is antisymmetric in $murho$, $nusigma$, and $lambda$ does not obey any symmetries?




tensors index-notation young-tableaux




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    up vote
    1
    down vote

    favorite












    I'm reading a paper which has given the index symmetries in terms of a Young Tableau which I'm having trouble understanding e.g. one is of the form



    $[mu][nu]$

    $[rho][sigma]$



    I understand that if it was fully horizontal (vertical) then it would correspond to a totally (anti)symmetric tensor in all its indices. In this example I'm given to believe it is actually ambiguous in that it could correspond to a tensor which is either symmetric in exchange of $munu$ and $rhosigma$ i.e. in the bracket notation then it'd be $T^munurhosigma$ = $T^(munu)(rhosigma)$, but also in fact could be used to represent a tensor which is antisymmetric in $murho$ and $nusigma$ i.e. $T^murhonusigma$ = $T^[murho][nusigma]$. Is this the case or have I misunderstood?



    Also in a slightly more complicated case, say



    $[mu][nu][lambda]$

    $[rho][sigma]$



    Can this be read as a tensor symmetric in $munulambda$ and $rho sigma$, or a tensor which is antisymmetric in $murho$, $nusigma$, and $lambda$ does not obey any symmetries?




    tensors index-notation young-tableaux




    share|cite|improve this question





















      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I'm reading a paper which has given the index symmetries in terms of a Young Tableau which I'm having trouble understanding e.g. one is of the form



      $[mu][nu]$

      $[rho][sigma]$



      I understand that if it was fully horizontal (vertical) then it would correspond to a totally (anti)symmetric tensor in all its indices. In this example I'm given to believe it is actually ambiguous in that it could correspond to a tensor which is either symmetric in exchange of $munu$ and $rhosigma$ i.e. in the bracket notation then it'd be $T^munurhosigma$ = $T^(munu)(rhosigma)$, but also in fact could be used to represent a tensor which is antisymmetric in $murho$ and $nusigma$ i.e. $T^murhonusigma$ = $T^[murho][nusigma]$. Is this the case or have I misunderstood?



      Also in a slightly more complicated case, say



      $[mu][nu][lambda]$

      $[rho][sigma]$



      Can this be read as a tensor symmetric in $munulambda$ and $rho sigma$, or a tensor which is antisymmetric in $murho$, $nusigma$, and $lambda$ does not obey any symmetries?




      tensors index-notation young-tableaux




      share|cite|improve this question











      I'm reading a paper which has given the index symmetries in terms of a Young Tableau which I'm having trouble understanding e.g. one is of the form



      $[mu][nu]$

      $[rho][sigma]$



      I understand that if it was fully horizontal (vertical) then it would correspond to a totally (anti)symmetric tensor in all its indices. In this example I'm given to believe it is actually ambiguous in that it could correspond to a tensor which is either symmetric in exchange of $munu$ and $rhosigma$ i.e. in the bracket notation then it'd be $T^munurhosigma$ = $T^(munu)(rhosigma)$, but also in fact could be used to represent a tensor which is antisymmetric in $murho$ and $nusigma$ i.e. $T^murhonusigma$ = $T^[murho][nusigma]$. Is this the case or have I misunderstood?



      Also in a slightly more complicated case, say



      $[mu][nu][lambda]$

      $[rho][sigma]$



      Can this be read as a tensor symmetric in $munulambda$ and $rho sigma$, or a tensor which is antisymmetric in $murho$, $nusigma$, and $lambda$ does not obey any symmetries?





      tensors index-notation young-tableaux





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      share|cite|improve this question




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      asked Aug 6 at 14:20









      dknt

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