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Consider the simple Lie algebra $A_{n-1}$ over the complex numbers. I refer to $A_{n-1}$ rather than $A_n$ for reasons that I try to make clearer below. The following facts are no doubt well known, but I will state them here for convenience:

  • Every representation of $A_{n-1}$ can be indexed by its highest weight, $\Lambda$, and such a weight is a linear combination of the fundamental weights: $\Lambda = a_1 \omega_1 + \cdots + a_{n-1}\omega_{n-1}$.
  • Given an irreducible representation $\Gamma_\Lambda$ with $\Lambda$ as above, we get a tuple of length $n$: $(a_1, \dots, a_{n-1}, 0)$. Then the dimension of this representation may be calculated by the following dimension formula from Fulton and Harris: $$ \dim \Gamma_\Lambda = \prod_{1 \leq i < j \leq n} {(a_i + \cdots + a_{j-1}) + j - i \over j - i} $$

I have been trying to prove that the only irreducible representations of $A_{n-1}$ with dimension equal to $p^2$ are the representations $\Gamma_{(p^2 - 1) \cdot \omega_1}$ of $A_1$ and $\Gamma_{\omega_1}, \Gamma_{\omega_{p^2 - 1}}$ of $A_{p^2 - 1}$. I am still a ways off from proving this, (I have reason to believe it is true from some computations on Sage), but as a start I have been trying to bound the dimensions of the fundamental representations of $A_{n-1}$ i.e., those representations for which $\Lambda = k\cdot\omega_i$, $k \in \Bbb N$, $1 \leq i \leq n-1$.

The tuple such a representation corresponds to is the following: $(0, \dots, 0, k, 0, \dots, 0)$ where $k$ is in the $i$th position in the tuple of length $n$ (again, this corresponds to a representation of $A_{n-1}$). Based on my calculations for $i = 1,2,3$ i.e., $k$ in the first, second, and third slot respectively, it looks as though, for a representation $\Gamma_{k \cdot \omega_i}$ of $A_{n-1}$, that the dimension is given by the following combinatorial formula: $$ d(n, k, i) = {\prod_{\xi = 1}^i {n + k - \xi \choose k } \over \prod_{\chi = 1}^{i-1} { k + \chi \choose k}} $$

Please note that I have not proved that this formula is in fact correct, it is simply my conjecture at this point. This brings me to my question:

If true, such a formula for the dimension of the fundamental representations is quite neat and tidy. Therefore, it is believable that I have simply rederived something previously done. Is this the case and, if so, please refer me to that derivation.

Many thanks for your time.

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1 Answer 1

This is Weyl's dimension formula. The dimensions of the fundamental modules $L(\omega_1), \ldots ,L(\omega_{n-1})$ for $A_{n-1}$ are $\binom{n}{1}, \binom{n}{2},\ldots ,\binom{n}{n-1}$. The dimensions for $L(k\omega_i)$ can also be computed from Weyl's dimension formula. This is done (in part) in Knapp's book Lie groups beyond an introduction, Theorem $5.84$ and the examples thereafter.
There is also a "universal" dimension formula for the dimension of the Cartan powers of the adjoint representation of a complex simple Lie algebra, independent of the type. See here: http://www.math.tamu.edu/~jml/LMunivpub.pdf. I hope this "derivation" helps.

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