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Math Help - Specht Modules, Row- and Column-Stabilisers

  1. #1
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    Specht Modules, Row- and Column-Stabilisers

    Reading The Symmetric Group by Bruce Sagan and struggling with a couple of trivial definitions - where I've had trouble I've written (WHY).

    If we have a tableau t with rows R_i and columns C_j, then we define the row- and column-stabilisers respectively to be

    R_t:=\Pi S_{R_i}, \quad C_t:=\Pi S_{C_j}.

    Apparently then equivalence classes of tableaux {t} (called tabloids, i.e. where the order of numbers in rows is immaterial) are given by  \{t\} = R_t t (WHY).

    In addition, these groups are associated with certain elements of \mathbb{C}[S_n]. In general, given a subset H\subseteq S_n we can form the group algebra sums  H^+:=\sum_{\pi\in H} \pi,\;\; H^-:=\sum_{\pi\in H}\textrm{sign}(\pi)\pi.

    For a tableau t, the element  R_t^+ is already implicit in the corresponding tabloid by the remark above. (WHY. Don't understand this at all - isn't R_t^+ going to be an ordered tuple? What does it look like? This line makes no sense to me at all).

    We also need to make use of

    \kappa_t:=C_t^-.

    Note that if t has columns C_j then \kappa_t factors as \kappa_t=\Pi\kappa_{C_j} (WHY - how can a k-tuple (assuming there are k columns) be expressed as a product of single numbers? And what does the sum of permutations mean anyway? Very confused here).

    Any light anyone could shed on any of the above would be great, I'm having a bit of trouble digesting it all. This is a great book though.

    Cheers,

    Atticus
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  2. #2
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    Quote Originally Posted by AtticusRyan View Post
    Reading The Symmetric Group by Bruce Sagan and struggling with a couple of trivial definitions - where I've had trouble I've written (WHY).

    If we have a tableau t with rows R_i and columns C_j, then we define the row- and column-stabilisers respectively to be

    R_t:=\Pi S_{R_i}, \quad C_t:=\Pi S_{C_j}.

    Apparently then equivalence classes of tableaux {t} (called tabloids, i.e. where the order of numbers in rows is immaterial) are given by  \{t\} = R_t t (WHY).
    A tabloid {t} is a set of tableaux t_i generated by row permutations applied to t_i.

    For instance, let \lambda \vdash n where \lambda=(2,1), n=3, and its first row consists of 1,2 and its second row consists of 3. Then {t} = {t_1, t_2} where t_1 has its first row 1,2 and its second row 3, and t_2 has its first row 2,1 and its second row 3.
    It is becauase that a tabloid is equivalent up to permutations in the row elements of the given tableau.

    Consider another "tableau (not tabloid)" whose first row consists of 2,3,4 and second row consists of 1, 5. It has R_t = S_{\{2,3,4\}} \times S_{\{1,5\}}. So, (2,3), (3,4), (2,3)(1,5), (2,3,4)(1,5),.. are all elements of R_t, denoted R_t={ (2,3), (3,4), (2,3)(1,5), (2,3,4)(1,5),.. }. When we consider {t} instead of t, {t} consists of all those tableaux generated by row permutations. For instance, if we apply (2,3)t, then (2,3)t has its first row 3,2,4 and its second row 1,5 (We should take an order into account because we are talking about a "tableau" here instead of a "tabloid"). Now (2,3)t is also a memeber of {t}. We can see that its general form has  \{t\} = R_t t.

     \{t\} = R_t t
    In addition, these groups are associated with certain elements of \mathbb{C}[S_n]. In general, given a subset H\subseteq S_n we can form the group algebra sums  H^+:=\sum_{\pi\in H} \pi,\;\; H^-:=\sum_{\pi\in H}\textrm{sign}(\pi)\pi.
    For a tableau t, the element  R_t^+ is already implicit in the corresponding tabloid by the remark above. (WHY. Don't understand this at all - isn't R_t^+ going to be an ordered tuple? What does it look like? This line makes no sense to me at all).
    R_t^{+} looks like (2,3)+ (3,4)+(2,3)(1,5)+ (2,3,4)(1,5),.. using our previous tableau. It corresponds to a_{\lambda} in the link.
    \kappa_t:=C_t^-.

    Note that if t has columns C_j then \kappa_t factors as \kappa_t=\Pi\kappa_{C_j} (WHY - how can a k-tuple (assuming there are k columns) be expressed as a product of single numbers? And what does the sum of permutations mean anyway? Very confused here).Any light anyone could shed on any of the above would be great, I'm having a bit of trouble digesting it all. This is a great book though.

    Cheers,

    Atticus
    This definition is used to construct polytabloids and has a good example in your Sagan's textbook p61 after definition 2.3.2.
    You will see that polytabloids of standard \lambda-tableaux are a basis of the Specht module of shape \lambda.
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