Closed formulas for the character of the symmetric group

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I know the Murnaghan–Nakayama rule, but I am wondering if there is any closed formulas for the character of the symmetric group. I know the following:



$$chi_n(sigma) = 1$$
$$chi_11...1(sigma) = sgn(sigma)$$
$$chi_n-1,1(sigma) = fix(sigma)-1$$
$$chi_21...1(sigma) = sgn(sigma)(fix(sigma) - 1)$$



Are they any other simple formulas like these? I know that the answer is no for the general case, but maybe there is in simple cases, like for the other hook partitions or for rectangle partition?



Thanks in advance!



Étienne






rt.representation-theory symmetric-groups characters







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    up vote
    2
    down vote

    favorite












    I know the Murnaghan–Nakayama rule, but I am wondering if there is any closed formulas for the character of the symmetric group. I know the following:



    $$chi_n(sigma) = 1$$
    $$chi_11...1(sigma) = sgn(sigma)$$
    $$chi_n-1,1(sigma) = fix(sigma)-1$$
    $$chi_21...1(sigma) = sgn(sigma)(fix(sigma) - 1)$$



    Are they any other simple formulas like these? I know that the answer is no for the general case, but maybe there is in simple cases, like for the other hook partitions or for rectangle partition?



    Thanks in advance!



    Étienne






    rt.representation-theory symmetric-groups characters







    share|cite|improve this question

























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      I know the Murnaghan–Nakayama rule, but I am wondering if there is any closed formulas for the character of the symmetric group. I know the following:



      $$chi_n(sigma) = 1$$
      $$chi_11...1(sigma) = sgn(sigma)$$
      $$chi_n-1,1(sigma) = fix(sigma)-1$$
      $$chi_21...1(sigma) = sgn(sigma)(fix(sigma) - 1)$$



      Are they any other simple formulas like these? I know that the answer is no for the general case, but maybe there is in simple cases, like for the other hook partitions or for rectangle partition?



      Thanks in advance!



      Étienne






      rt.representation-theory symmetric-groups characters







      share|cite|improve this question















      I know the Murnaghan–Nakayama rule, but I am wondering if there is any closed formulas for the character of the symmetric group. I know the following:



      $$chi_n(sigma) = 1$$
      $$chi_11...1(sigma) = sgn(sigma)$$
      $$chi_n-1,1(sigma) = fix(sigma)-1$$
      $$chi_21...1(sigma) = sgn(sigma)(fix(sigma) - 1)$$



      Are they any other simple formulas like these? I know that the answer is no for the general case, but maybe there is in simple cases, like for the other hook partitions or for rectangle partition?



      Thanks in advance!



      Étienne






      rt.representation-theory symmetric-groups characters




      rt.representation-theory symmetric-groups characters






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      eti902

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          2 Answers
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          Giving explicit formulas for the characters is the content of the recent article "An explicit formula for the characters of the symmetric group" by
          Michel Lassalle :
          https://link.springer.com/article/10.1007/s00208-007-0156-5.






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            For generalizing the formulas in your question, see http://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i2r19 and Examples 1.7.13 and 1.7.14 in Macdonald's Symmetric Functions and Hall Polynomials, 2nd ed.
            For a different formula, see https://www.researchgate.net/publication/227299451_Stanley%27s_Formula_for_Characters_of_the_Symmetric_Group.






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              2 Answers
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              active

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              up vote
              4
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              accepted










              Giving explicit formulas for the characters is the content of the recent article "An explicit formula for the characters of the symmetric group" by
              Michel Lassalle :
              https://link.springer.com/article/10.1007/s00208-007-0156-5.






              share|cite|improve this answer
























                up vote
                4
                down vote



                accepted










                Giving explicit formulas for the characters is the content of the recent article "An explicit formula for the characters of the symmetric group" by
                Michel Lassalle :
                https://link.springer.com/article/10.1007/s00208-007-0156-5.






                share|cite|improve this answer






















                  up vote
                  4
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                  accepted







                  up vote
                  4
                  down vote



                  accepted






                  Giving explicit formulas for the characters is the content of the recent article "An explicit formula for the characters of the symmetric group" by
                  Michel Lassalle :
                  https://link.springer.com/article/10.1007/s00208-007-0156-5.






                  share|cite|improve this answer












                  Giving explicit formulas for the characters is the content of the recent article "An explicit formula for the characters of the symmetric group" by
                  Michel Lassalle :
                  https://link.springer.com/article/10.1007/s00208-007-0156-5.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 1 hour ago









                  Mare

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                      For generalizing the formulas in your question, see http://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i2r19 and Examples 1.7.13 and 1.7.14 in Macdonald's Symmetric Functions and Hall Polynomials, 2nd ed.
                      For a different formula, see https://www.researchgate.net/publication/227299451_Stanley%27s_Formula_for_Characters_of_the_Symmetric_Group.






                      share|cite
























                        up vote
                        1
                        down vote













                        For generalizing the formulas in your question, see http://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i2r19 and Examples 1.7.13 and 1.7.14 in Macdonald's Symmetric Functions and Hall Polynomials, 2nd ed.
                        For a different formula, see https://www.researchgate.net/publication/227299451_Stanley%27s_Formula_for_Characters_of_the_Symmetric_Group.






                        share|cite






















                          up vote
                          1
                          down vote










                          up vote
                          1
                          down vote









                          For generalizing the formulas in your question, see http://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i2r19 and Examples 1.7.13 and 1.7.14 in Macdonald's Symmetric Functions and Hall Polynomials, 2nd ed.
                          For a different formula, see https://www.researchgate.net/publication/227299451_Stanley%27s_Formula_for_Characters_of_the_Symmetric_Group.






                          share|cite












                          For generalizing the formulas in your question, see http://www.combinatorics.org/ojs/index.php/eljc/article/view/v16i2r19 and Examples 1.7.13 and 1.7.14 in Macdonald's Symmetric Functions and Hall Polynomials, 2nd ed.
                          For a different formula, see https://www.researchgate.net/publication/227299451_Stanley%27s_Formula_for_Characters_of_the_Symmetric_Group.







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                          answered 5 mins ago









                          Richard Stanley

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