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This might be a classic question, but since I am new to representation theory of the symmetric group, I am asking it here.

Suppose that $\lambda_1 \geq \lambda_2 \geq \dots \lambda_k$ and $\rho$ be the irreducible representation of $S_n$ associated with Young tableau $(\lambda_1,\dots,\lambda_k)$ where $n=\sum \lambda_i$. What can be said about dimension of $\rho$? I am specifically interested in the case where we have two rows (i.e. $k=2$) and the case where $\lambda_i =1$ for $i\geq 3$.

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    $\begingroup$ This is not a research level question. You should use MSE for questions like that. $\endgroup$
    – Vladimir Dotsenko
    Commented Dec 6, 2017 at 12:44
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    $\begingroup$ @VladimirDotsenko we all have to start somewhere, and this strikes me as a question that a researcher in Area One may encounter that is basic/easy for a researcher in Area Two $\endgroup$
    – Yemon Choi
    Commented Dec 6, 2017 at 13:49
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    $\begingroup$ @YemonChoi I think that a topic that is discussed in nearly every (non-advanced) course on representation theory is clearly not a research level question. Frankly, I am puzzled why the OP did not bother to open a representation theory textbook first. $\endgroup$
    – Vladimir Dotsenko
    Commented Dec 6, 2017 at 15:56
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    $\begingroup$ @VladimirDotsenko Because when you're not in the field, you don't know what is easy. $\endgroup$
    – David E Speyer
    Commented Dec 6, 2017 at 19:21
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    $\begingroup$ Actually, it is rather bizarre that the OP knew that there exists an irreducible representation associated with a Young Tableau, and however he did not know the hook length formula. Usually, the two things go (almost) together, at least in the literature I'm familiar with. $\endgroup$
    – Francesco Polizzi
    Commented Dec 6, 2017 at 19:30

1 Answer 1

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This is a classical application of Frobenius formula and Vandermonde determinant. We have $$\dim V_{\lambda}=\frac{n!}{l_1! \cdots l_k!} \prod_{i<j}(l_i-l_j),$$ where $l_i=\lambda_i+k-i$. See Section 4.1 of

W. Fulton, J. Harris: Representation Theory (a first course), GTM 129 (1991).

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    $\begingroup$ maybe one should also mention the term "Hook length formula "... $\endgroup$
    – Frieder Ladisch
    Commented Dec 6, 2017 at 15:43
  • $\begingroup$ Right. It is contained in the given reference. $\endgroup$
    – Francesco Polizzi
    Commented Dec 6, 2017 at 17:14

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