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For any simple complex Lie algebra $\frak{g}$, with a given choice of Cartan subalgebra $\frak{h}$, we have an associated root system $R \subseteq \frak{h}^*$. The properties of $R$ can be formalized as an abstract root system allowing for a classification of simple complex Lie algebras.

For any irreducible representation $V$ of $\frak{g}$ we have an associated finite set of weight vectors, which again live in $\frak{h}^*$. These do not form a root system since they are not closed under multiplication by $-1$. (I also guess they are not closed under the action of the Weyl group of the relections in weights, but I am not sure.)

Is there some abstract system, generalizing root systems that abstracts the properties of weights? If so, how does it reduce to a root sysmtem for the special case of the adjoint representation?

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    $\begingroup$ Well the weights of an irrep are the lattice points in a $W$-permutohedron. We could state what that means in a purely polytopal way. $\endgroup$
    – Sam Hopkins
    Commented Sep 4, 2021 at 17:59
  • $\begingroup$ Wait what? Not closed under multiplication by $-1$? I always thought they were! Can you give an example of where they are not? $\endgroup$
    – Vincent
    Commented Sep 7, 2021 at 13:26
  • $\begingroup$ @Vincent: $-1$ need not be an element of the Weyl group (for example in $\mathfrak{sl}_2$ it is not). $\endgroup$
    – Sam Hopkins
    Commented Sep 7, 2021 at 13:27
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    $\begingroup$ Sorry I meant $\mathfrak{sl}_3$. $\endgroup$
    – Sam Hopkins
    Commented Sep 7, 2021 at 13:29
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    $\begingroup$ Aaah, right, now I see the issue. The weights are still, in an informal sense, spread out evenly around zero, and not, say, shifted to one side of the lattice, but that in itself need not mean that they are invariant under multiplication with $-1$ since they can form e.g. an equilateral triangle with 0 as the center of mass. Right. Thank you! $\endgroup$
    – Vincent
    Commented Sep 7, 2021 at 13:35

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If $\Phi$ is a(n abstract) root system in a Euclidean vector space $V$, then we have notions of simple roots $\alpha_1,\ldots,\alpha_r$, fundamental weights $\omega_1,\ldots,\omega_r$, root lattice $Q := \mathrm{Span}_{\mathbb{Z}}\{\alpha_1,\ldots,\alpha_r\}$, weight lattice $P := \mathrm{Span}_{\mathbb{Z}}\{\omega_1,\ldots,\omega_r\}$, et cetera, for it purely in linear algebraic terms (without any reference to a Lie algebra).

If $W$ is the Weyl group of $\Phi$, then for a (dominant, integral) weight $\lambda$ (i.e., nonnegative combination of the $\omega_i$), the set of weights of the irreducible representation $V^\lambda$ is the same as $\mathrm{ConvHull}(W\lambda) \cap (Q+\lambda)$. This gives you a purely convex polytopal description of the set of weights of an irrep. It is possible to do more here as well: e.g., describe what are the vertices of the polytope, or what are its facets. But maybe it is not exactly what you are looking for.

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    $\begingroup$ The point is: everything about weights can be defined just in terms of a root system, without talking about Lie things at all. $\endgroup$
    – Sam Hopkins
    Commented Sep 7, 2021 at 13:33

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