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Young tableux (representation theory)

  1. Jan 4, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider the irreducible representation [itex]V[/itex] in the symmetric group [itex]S_5[/itex] corresponding to the Young diagram (these are meant to be boxes): [tex][\;\;][\;\;] \\ [\;\;][\;\;] \\ [\;\;]\;\;\;\;[/tex]

    (a) List all standard Young tableaux of the given shape (that is, list all the possible fillings of the boxes with $\{1,2,3,4,5\}$ such that the numbers increase along the rows and down the columns).

    (b) Give the dimension of [itex]V[/itex].

    (c) For any standard tableau [itex]\Omega[/itex] of the given shape, let [itex]e_{\Omega}[/itex] be the standard basis element of the representation [itex]V[/itex]. Evaluate the action of the permutation [itex]g=(23)(45)(23)(34)(23) \in S_5[/itex] on the vector [itex]e_{\Omega}[/itex] where [itex]\Omega =[/itex] [tex][\;1\;][\;4\;] \\ [\;2\;][\;5\;] \\ [\;3\;]\;\;\;\;\;\,[/tex]
    2. Relevant equations



    3. The attempt at a solution

    My answers:
    (a) There are 5 standard Young tableaux of the given shape.
    (b) So dim(V)=5.
    (c) The basis vector [itex]e_{\Omega} = P_{\Omega}Q_{\Omega}[/itex] where [itex]P_\Omega \in \mathbb{C}S_5[/itex] and [itex]Q_{\Omega}\in\mathbb{C}S_5[/itex] are the row symmetrizer and column antisymmetrizer of [itex]\Omega[/itex]: [tex]P_{\Omega} = [e+(14)][e+(25)] = e + (25) + (14) + (14)(25) \\ Q_{\Omega} = [e - (12) - (13) - (23) + (123) + (132)][e-(45)][/tex]

    The given permutation [itex]g=(23)(45)(23)(34)(23)=(2543)[/itex]. Is there a quicker way to evaluate the action of this permutation on [itex]e_{\Omega}[/itex] then to work out [itex](2543)P_{\Omega}Q_{\Omega}[/itex]?
     
    Last edited by a moderator: Jan 4, 2013
  2. jcsd
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