Source code for my notes on representations of semisimple Lie algebras and Olivier Mathieu's classification of simple weight modules
Minor tweak in notation
Changed the notation for the result which establishes a correspodance between sl(n)-sequences and weights of sl(n)
1 files changed, 10 insertions, 10 deletions
| Status | Name | Changes | Insertions | Deletions |
| Modified | sections/coherent-families.tex | 2 files changed | 10 | 10 |
diff --git a/sections/coherent-families.tex b/sections/coherent-families.tex @@ -210,7 +210,8 @@ Example~\ref{ex:cartan-of-sl} and consider the linear functionals \(\epsilon_i, \begin{proposition} The map \begin{align*} - m : \mathfrak{h}^* & \to K^n \\ + m : \mathfrak{h}^* & + \to \{ \mathfrak{sl}_n\textrm{\normalfont-sequences} \} \\ \lambda & \mapsto ( @@ -219,15 +220,14 @@ Example~\ref{ex:cartan-of-sl} and consider the linear functionals \(\epsilon_i, \kappa(\epsilon_n, \lambda + \rho) ) \end{align*} - is \(W\)-equivariant bijection onto the space of all - \(\mathfrak{sl}_n\)-sequences, where the action \(W \cong S_n\) on - \(\mathfrak{h}^*\) is given by the dot-action and the action of \(W\) on - \(K^n\) is given my permuting coordinates. A weight \(\lambda \in - \mathfrak{h}^*\) satisfies the conditions of - Lemma~\ref{thm:sl-bounded-weights} if, and only if - the diferences between all but one consecutive coordinates of \(m(\lambda)\) - are positive integers -- i.e. there is some unique \(i < n\) such that - \(m(\lambda)_i - m(\lambda)_{i + 1}\) is \emph{not} a positive integer. + is \(W\)-equivariant bijection, where the action \(W \cong S_n\) on + \(\mathfrak{h}^*\) is given by the dot-action and the action of \(W\) on the + space of \(\mathfrak{sl}_n\)-sequences is given my permuting coordinates. A + weight \(\lambda \in \mathfrak{h}^*\) satisfies the conditions of + Lemma~\ref{thm:sl-bounded-weights} if, and only if the diferences between all + but one consecutive coordinates of \(m(\lambda)\) are positive integers -- + i.e. there is some unique \(i < n\) such that \(m(\lambda)_i - m(\lambda)_{i + + 1}\) is \emph{not} a positive integer. \end{proposition} % TODO: Change the notation for 𝓟 ? The paper on affine vertex algebras calls