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I've been studying representation theory of symmetric group on Tung's Group Theory in Physics. I understood that different Young Diagrams corresponds to inequivalent irreducible representations of symmetric group (Theorem 5.6). In fact, they exhaust all irreducible representations (Theorem 5.7) and distinct standard Young Tableaux are linearly independent and the direct sum of left ideals generated by them span the whole group algebra space (Theorem 5.8).

Apologizing for my ignorance, but I have two questions that I cannot figure out straightforwardly following the derivations and arguments provided in the textbook:

(1) It can be shown (Theorem 5.5) that any permutation on a Young Tableaux gives an equivalent representation, so that $e_\lambda^p \equiv p e_\lambda p^{-1}$ is equivalent to $e_\lambda$. Since $e_\lambda^p$ is not necessarily a standard Young Tableaux, while it is an idempotent (since it is equivalent to an idempotent), the left ideal generated by it should be coincided with one of those generated by a standard Young Tableaux?

(2) Essentially primitive idempotent is defined (with more derivations given in the appendix IV of the textbook) as $e_\lambda \equiv s_\lambda a_\lambda$. I am wondering, by employing similar arguments used in the appendix (to be specific, one exchanges the line with column, symmetrizer with anti-symmetrizer in the arguments), it occurs to me that $\tilde{e}_\lambda \equiv a_\lambda s_\lambda$ is also essentially primitive idempotent. If I am wrong, which important piece I have missed, if not, will its corresponding left ideal be also coincided with some standard Young Tableaux ?

If I stray too far away from the right track, please delete this question. Many thanks.

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  • $\begingroup$ Concerning the second question, it seems to me that $\tilde{e}_\lambda e_\lambda=s_\lambda a_\lambda a_\lambda s_\lambda =\eta s_\lambda a_\lambda s_\lambda = e_\lambda s_\lambda \ne 0$, so $\tilde{e}_\lambda$ is equivalent to $e_\lambda$. But the question still remains: does it coincide with some left ideal generated by a specific standard $e_\lambda^p$ (standard Young Tableaux)? $\endgroup$ – gamebm May 17 '15 at 12:41
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    $\begingroup$ (1) Sure it can coincide. If $p$ is a vertical permutation of $\Theta_\lambda$, they will coincide, because the left ideal generated by $p e_\lambda p^{-1}$ is the left ideal generated by $e_\lambda p^{-1}$ which, in this case, is the left ideal generated by $e_\lambda$. $\endgroup$ – darij grinberg May 19 '15 at 3:03
  • $\begingroup$ Question (2) is a good one. Do we know that the left ideal generated by $\widetilde{e}_\lambda$ is irreducible to begin with? $\endgroup$ – darij grinberg May 19 '15 at 3:04
  • $\begingroup$ Thx for the comments! (1) Yes, but in general, since vertical permutation does not simply get to any tableaux, do we have some arguments for just any given tableaux. (I understand since irreducible left ideals coincide with or be distinct from each other, so one common element does the trick). (2) Can we employ similar arguments leaded to those of $e_\lambda$, replacing all symmetrizer by antisymmetrizer, row by column? $\endgroup$ – gamebm May 19 '15 at 10:05
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    $\begingroup$ (2) See math.stackexchange.com/questions/471223/… . $\endgroup$ – darij grinberg Aug 14 '15 at 21:40

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