lie-algebras-and-their-representations

diff --git a/sections/fin-dim-simple.tex b/sections/fin-dim-simple.tex
@@ -481,23 +481,24 @@ called \emph{basis}.
 \begin{definition}\label{def:basis-of-root}\index{weights!basis}
   A subset \(\Sigma = \{\beta_1, \ldots, \beta_r\} \subset \Delta\) of linearly
   independent roots is called \emph{a basis for \(\Delta\)} if, given \(\alpha
-  \in \Delta\), there are \(k_1, \ldots, k_r \in \mathbb{N}\) such that
+  \in \Delta\), there are unique \(k_1, \ldots, k_r \in \mathbb{N}\) such that
   \(\alpha = \pm(k_1 \beta_1 + \cdots + k_r \beta_r)\).
 \end{definition}
 
 The interesting thing about basis for \(\Delta\) is that they allow us to
 compare weights of a given \(\mathfrak{g}\)-module. At this point the reader
 should be asking himself: how? Definition~\ref{def:basis-of-root} isn't exactly
-all that intuitive. Well, the thing is that any choice of basis induces a
-partial order in \(Q\), where elements are ordered by their \emph{heights}.
+all that intuitive. Well, the thing is that any choice of basis \(\Sigma\)
+induces an order in \(Q\), where elements are ordered by their
+\emph{\(\Sigma\)-coordinates}.
 
 \begin{definition}\index{weights!orderings of roots}
   Let \(\Sigma = \{\beta_1, \ldots, \beta_r\}\) be a basis for \(\Delta\).
   Given \(\alpha = k_1 \beta_1 + \cdots + k_r \beta_r \in Q\) with \(k_1,
-  \ldots, k_r \in \mathbb{Z}\), we call the number \(\operatorname{ht}(\alpha)
-  = k_1 + \cdots + k_r \in \mathbb{Z}\) \emph{the height of \(\alpha\)}. We say
-  that \(\alpha \preceq \beta\) if \(\operatorname{ht}(\alpha) \le
-  \operatorname{ht}(\beta)\).
+  \ldots, k_r \in \mathbb{Z}\), we call the vector \(\alpha_\Sigma = (k_1,
+  \ldots, k_r) \in \mathbb{Z}^r\) \emph{the \(\Sigma\)-coordinate of
+  \(\alpha\)}. We say that \(\alpha \preceq \beta\) if \(\alpha_\Sigma \le
+  \beta_\Sigma\) in the lexicographical order.
 \end{definition}
 
 \begin{definition}