Permutation of an element












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An array $mathbb T$ has elements $T_{ijkl}$ where $i,j,k,l=1,2,3,4$. It is given that
$$T_{ijkl}=T_{jikl}=T_{ijlk}=-T_{klij}$$
for all values of $i,j,k,l$. The number of independent components in this array is




I don't know how to solve this question, please help. I tried thinking about all the permutation of this element but can't figure out how to think about the condition given here.



Sorry for my bad english, this is my first question.






permutations







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    0












    $begingroup$



    An array $mathbb T$ has elements $T_{ijkl}$ where $i,j,k,l=1,2,3,4$. It is given that
    $$T_{ijkl}=T_{jikl}=T_{ijlk}=-T_{klij}$$
    for all values of $i,j,k,l$. The number of independent components in this array is




    I don't know how to solve this question, please help. I tried thinking about all the permutation of this element but can't figure out how to think about the condition given here.



    Sorry for my bad english, this is my first question.






    permutations







    share|cite|improve this question











    $endgroup$















      0












      0








      0





      $begingroup$



      An array $mathbb T$ has elements $T_{ijkl}$ where $i,j,k,l=1,2,3,4$. It is given that
      $$T_{ijkl}=T_{jikl}=T_{ijlk}=-T_{klij}$$
      for all values of $i,j,k,l$. The number of independent components in this array is




      I don't know how to solve this question, please help. I tried thinking about all the permutation of this element but can't figure out how to think about the condition given here.



      Sorry for my bad english, this is my first question.






      permutations







      share|cite|improve this question











      $endgroup$





      An array $mathbb T$ has elements $T_{ijkl}$ where $i,j,k,l=1,2,3,4$. It is given that
      $$T_{ijkl}=T_{jikl}=T_{ijlk}=-T_{klij}$$
      for all values of $i,j,k,l$. The number of independent components in this array is




      I don't know how to solve this question, please help. I tried thinking about all the permutation of this element but can't figure out how to think about the condition given here.



      Sorry for my bad english, this is my first question.






      permutations




      permutations






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Jan 17 at 5:06









      Daniel Mathias

      1,30518




      1,30518










      asked Jan 17 at 4:13









      Harshit TiwariHarshit Tiwari

      32




      32






















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          $begingroup$

          You are given that your tensor $T_{ijkl}$ is symmetric on first pair of indices, on last pair of indices and antisymmetric on pair switch.



          If tensor is symmetric on pair of indices, the order in this pair doesn't matter, so we can count them as a set. Set of 2 indices
          $$
          {i,j} ={1,1}, {1,2}, {2,2}, {1,3},ldots{4,4}
          $$
          in total $k={nchoose 2}+n={4choose 2}+4=10$ elements (we sum the number of pairs with different element with the number of pairs where elements are equal).



          If there wasn't a last condition on $T_{ijkl}=-T_{klij}$, then the total number of independent elements would be $10times10=100$. However, because of the condition, we need to subtract $k=10$ elements, since for each ${i,j}={k,l}$, $T_{ijkl}=T_{klij}=0$, and then divide by 2. So the total number of independent elements is $(10^2-10)/2=45$






          share|cite|improve this answer









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            $begingroup$

            You are given that your tensor $T_{ijkl}$ is symmetric on first pair of indices, on last pair of indices and antisymmetric on pair switch.



            If tensor is symmetric on pair of indices, the order in this pair doesn't matter, so we can count them as a set. Set of 2 indices
            $$
            {i,j} ={1,1}, {1,2}, {2,2}, {1,3},ldots{4,4}
            $$
            in total $k={nchoose 2}+n={4choose 2}+4=10$ elements (we sum the number of pairs with different element with the number of pairs where elements are equal).



            If there wasn't a last condition on $T_{ijkl}=-T_{klij}$, then the total number of independent elements would be $10times10=100$. However, because of the condition, we need to subtract $k=10$ elements, since for each ${i,j}={k,l}$, $T_{ijkl}=T_{klij}=0$, and then divide by 2. So the total number of independent elements is $(10^2-10)/2=45$






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              You are given that your tensor $T_{ijkl}$ is symmetric on first pair of indices, on last pair of indices and antisymmetric on pair switch.



              If tensor is symmetric on pair of indices, the order in this pair doesn't matter, so we can count them as a set. Set of 2 indices
              $$
              {i,j} ={1,1}, {1,2}, {2,2}, {1,3},ldots{4,4}
              $$
              in total $k={nchoose 2}+n={4choose 2}+4=10$ elements (we sum the number of pairs with different element with the number of pairs where elements are equal).



              If there wasn't a last condition on $T_{ijkl}=-T_{klij}$, then the total number of independent elements would be $10times10=100$. However, because of the condition, we need to subtract $k=10$ elements, since for each ${i,j}={k,l}$, $T_{ijkl}=T_{klij}=0$, and then divide by 2. So the total number of independent elements is $(10^2-10)/2=45$






              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$

                You are given that your tensor $T_{ijkl}$ is symmetric on first pair of indices, on last pair of indices and antisymmetric on pair switch.



                If tensor is symmetric on pair of indices, the order in this pair doesn't matter, so we can count them as a set. Set of 2 indices
                $$
                {i,j} ={1,1}, {1,2}, {2,2}, {1,3},ldots{4,4}
                $$
                in total $k={nchoose 2}+n={4choose 2}+4=10$ elements (we sum the number of pairs with different element with the number of pairs where elements are equal).



                If there wasn't a last condition on $T_{ijkl}=-T_{klij}$, then the total number of independent elements would be $10times10=100$. However, because of the condition, we need to subtract $k=10$ elements, since for each ${i,j}={k,l}$, $T_{ijkl}=T_{klij}=0$, and then divide by 2. So the total number of independent elements is $(10^2-10)/2=45$






                share|cite|improve this answer









                $endgroup$



                You are given that your tensor $T_{ijkl}$ is symmetric on first pair of indices, on last pair of indices and antisymmetric on pair switch.



                If tensor is symmetric on pair of indices, the order in this pair doesn't matter, so we can count them as a set. Set of 2 indices
                $$
                {i,j} ={1,1}, {1,2}, {2,2}, {1,3},ldots{4,4}
                $$
                in total $k={nchoose 2}+n={4choose 2}+4=10$ elements (we sum the number of pairs with different element with the number of pairs where elements are equal).



                If there wasn't a last condition on $T_{ijkl}=-T_{klij}$, then the total number of independent elements would be $10times10=100$. However, because of the condition, we need to subtract $k=10$ elements, since for each ${i,j}={k,l}$, $T_{ijkl}=T_{klij}=0$, and then divide by 2. So the total number of independent elements is $(10^2-10)/2=45$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Jan 17 at 14:19









                Vasily MitchVasily Mitch

                2,3141311




                2,3141311






























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