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I would be glad if someone can help me understand the argument in the first paragraph of page 4 of this paper.

Especially I don't understand their first sentence,

"Using N bosons (fermions) distributed over m states, one can construct completely symmetric (antisymmetric) irreducible representations of the group U(m) associated with Young tableaux with N boxes in a row (column)"

(I am quite familiar with the Quantum Statistics concepts being alluded to but not so much with the Young-Tableux technology being used)

All I can see is that $U(n)$ can act on the space of $c_i$ and keep the operators defined in their equation 2.9 and 2.10 unchanged.

Also on page 4 is there a typo in Equation 2.13? It doesn't seem to follow from the line previous to it and the line previous to it makes no sense to me. I guess there should have been a "=" between the $\lambda^N$ and $exp$ in the line just before 2.13.

Even if I make the above "correction" I don't see how 2.13 follows from it.

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    Let V be the defining representation of U(m). The symmetric power S^N(V) describes N bosons and it is an irreducible representation; the exterior power \Lambda^N(V) describes N fermions and it is also an irreducible representation. The Young tableaux are ways to label these representations among all representations.2011-01-25
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    @Qiaochu: why not an answer?2011-01-25
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    I didn't know which part he was confused about, and I also didn't look at the rest of the paper.2011-01-25
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    @Qiaochu: I hate it when you ruin a perfectly good rhetorical question with a reasonable response.2011-01-25
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    @Willie Now I am worried! I at least didn't intend my question to be rhetorical.2011-01-26
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    @Anirbit: I think Willie was referring to his own question to Qiaochu, not to your question.2011-01-26

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