Monomial irreducible \(\mathfrak{sl}_n(\mathbb C)\)-modules
Let \(L\) be a finite dimensional complex simple Lie algebra and \(H\) a Cartan subalgebra of \(L\). Given integral dominant weight \(\lambda\), let \(L(\lambda)\) denote the finite dimensional irreducible representation of \(L\) with highest weight \(\lambda\). The aim of the article under review is to find an ordered basis for \(L(\lambda)\) with respect to which the action of \(L\) on \(L(\lambda)\) can be represented by matrices. To the aim the author gives a suitable basis for \(L(\lambda)\) for \(L = \mathfrak{sl}_n(\mathbb{C})\) and for \(\lambda\) monomial weights. A simple formula for the action of Chevalley generators for \(\mathfrak{sl}_n(\mathbb{C})\) on the basis elements is also given. To obtain such a basis the fact that \(L(\lambda)\) is isomorphic to a symmetry class of tensors (see, \textit{A. Madadi} and \textit{M. Shahryari} [Linear Multilinear Algebra, 56, 517--541 (2008; Zbl 1147.15024)]) plays a central role.
scientific article; zbMATH DE number 6108179
scientific article