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For the Lie algebra $\frak{sl}_{n+1}$ we denote its fundamental irreducible representations by $V(\pi_i)$, with $i=1, \dots, n$. Where can I find a table of the character formula (in other words a list of all the non-zero weight spaces of $V(\pi_i)$ together with multiplicities)?

P.S. From this answer

Fundamental representations and weight space dimension

it looks that the multiplicities are also 1. So my question simplifies to a table of the non-zero weight spaces.

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These are examples of so called minuscule weights which are defined through the property that all the weights are given by $w\lambda$ where $\lambda$ is the highest weight and $w$ is an element of the Weyl group. So, in the $\mathfrak{sl}_{n+1}$, to get all the weights in "the $\epsilon$ basis" you just take all possible permutations of coordinates of $\lambda$.

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  • $\begingroup$ Thanks for the answer, but I could use a little more detail. Does Weyl conjugate just mean the action of an element of the Weyl group on a weight? When you say take all permutations, what is it that you are permuting? $\endgroup$
    – Martim Pereir
    Commented Feb 3, 2022 at 19:21
  • $\begingroup$ @MartimPereir Sorry, conjugated is perhaps not the right word to use here. I've edited the answer to make it more clear. Please don't hesitate to ask further questions. $\endgroup$
    – Vít Tuček
    Commented Feb 3, 2022 at 19:25
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    $\begingroup$ The weight lattice of $\mathfrak{sl}_{n+1}$ is $\mathbb{Z}^{n+1}/\mathbb{Z}(1,1,\ldots,1)$ and $\pi_i = (1,1,\ldots,1,0,0,\ldots,0)$ (with $i$ $1$'s). The nonzero weights in $V^{\pi_i}$ are then, as the answerer says, the permutations of $\pi_i$, i.e., all $1$/$0$ vectors with $i$ $1$'s. They all have multiplicity one. $\endgroup$
    – Sam Hopkins
    Commented Feb 3, 2022 at 19:28

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