When given a finite dimensional complex Lie algebra $\mathfrak{g}$ that is also semisimple and a choice of Cartan subalgebra $\mathfrak{h}$ we may talk about its weight lattice $\Lambda_{W} $ in $\mathfrak{h}^{*}$. In order to study the representation theory of $\mathfrak{g}$ we introduce a total ordering on $\Lambda_{W} $ by means of defining a real linear map $l$ from $\Lambda_{W} \otimes_\mathbb{\mathbb{z}} {\mathbb{R}}$ to $\mathbb{R}$ and that is irrational with respect to $\Lambda_{W} $ (first question: could you please define what irrationality in this context means?)

Now my main problem is this:

In practice, say when $\text{dim}_\mathbb{C}\mathfrak{h}=2, $ what happens is that we draw a line in an illustration of $\mathfrak{h}^{*}_\mathbb{R}$ (the real span of the roots; $\mathfrak{h}^{*}_\mathbb{R}\subset\mathfrak{h}$ is a real inner product space with respect to the Killing form) on paper and in some way the line defines $l$ (second question: could you please explain the convention on how drawing a line defines $l$).

The most important/practical third question I want to ask is: given a drawn line could you use it to immediately tell which of two weights in $\Lambda_{W} $ is larger? A friend told me that it's just a matter of perpendicular distances (the plain definition for lines drawn on paper) from the line but I don't think this is true, especially since it seems to me that the arbitrariness of our illustration of $\mathfrak{h}^{*}$ should mean there ought to be no relation to a fixed notion of perpendicularity.

An answer just addressing the third problem is more than welcome.

Finally, I've tried to explain my understanding of the situation in full, I'd be very grateful if you could correct any misunderstandings you see in this.


1 Answer 1


Suppose that $\Lambda\subset R^n$ is a lattice, i.e. a rank $n$ free abelian subgroup whose real span is the entire $R^n$. A linear function $l: R^n\to R$ is irrational with respect to the lattice $\Lambda$ if its restriction to $\Lambda$ is a monomorphism. In other words, if $v_1,...,v_n$ are generators of the lattice, their images are rationally independent real numbers. Given this, one defines an order on the lattice by: $$ u< v \iff l(u)< l(v). $$ This also answers your second question: to compare two elements with respect to this order, just compute their images! Now, if you want to do this more geometrically, you can assume (by rescaling) that $l$ has unit norm. Therefore, it can be identified with the orthogonal projection to a certain 1-dimensional linear subspace $L$ (a line) in $R^n$. (Which, in turn, is identified with ${\mathbb R}$ by introducing a basis in this subspace.) Informally, you can think of the orthogonal projection to $L$ as the "height" above the hyperplane $H$ orthogonal to $L$. Then $l(v)$ can be understood as the "signed distance" to $H$ (not the ordinary distance your friend was using), just by the elementary linear algebra: The orthogonal projection of $v$ to $P$ is given by the formula $v\mapsto v - f(v)$, $f(v)\in L$. Then $d(v, P)= |v - (v-f(v))|=|f(v)|$. Thus, $f(v)=\pm d(v, P)$. Hope it helps.


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