10
$\begingroup$

I'm working through Fulton-Harris and I'm kind of "stuck" at the following question. I'm looking for representations of $S_d$, the symmetric group on $d$ letters via Young Tableaux. The question is: "Show that for general $d$, the standard representation $V$ of $S_d$ corresponds to the partition $d = (d-1)+1$."

When I look at the hint I see that they give a basis: $$v_j = \sum_{g(d)=j} e_g - \sum_{h(1)=j} e_h$$ for $j=2,...,d$. Is there any way I can "see" that this should be a basis corresponding to the partition? Because, right now, they just seem to appear from magic. The representation should be the image of $c_{(d-1,1)} = a_{(d-1,1)} \cdot b_{(d-1,1)}$ in the group algebra of $S_d$ (here $a_\lambda$ corresponds to the permutations of $S_d$ that preserves the rows of the young tableaux and $b_\lambda$ the columns).

I do see how one might try for smaller cases by simply multiplying each element by $c_\lambda$ ( The young symmetrizer) , but even for $d=4$, this becomes quite impractical. Are there any other ways to see that this "should be" a basis? In general, what can we say about a young diagram and the corresponding basis for its irreducible representation?

  • 0
    Try to explain what all this is about precisely. What is "this basis" for instance. What group is being represented? Formulating this way it is difficult to help you out.2012-03-10
  • 0
    Thanks for the input Mark, I tried to clarify it a bit. Is it easier to understand now?2012-03-10

1 Answers 1