Constructions of representations of rank two semisimple Lie algebras with distributive lattices
type: Item
instance of: scholarly article
title: Constructions of representations of rank two semisimple Lie algebras with distributive lattices
zbMATH Open document ID: 1115.05096
author: L. Wyatt II Alverson
author: Scott J. Lewis
author: Robert Pervine
author: Donnelly, Robert G.
published in: The Electronic Journal of Combinatorics
publication date: 2007-03-12T00:00:00Z
full work available at URL: https://eudml.org/doc/127630
full work available at URL: v13i1r109.html
review text: Summary: We associate one or two posets (which we call ``semistandard posets'') to any given irreducible representation of a rank two semisimple Lie algebra over \(\mathbb{C}\). Elsewhere we have shown how the distributive lattices of order ideals taken from semistandard posets (we call these ``semistandard lattices'') can be used to obtain certain information about these irreducible representations. Here we show that some of these semistandard lattices can be used to present explicit actions of Lie algebra generators on weight bases (Theorem 5.1), which implies these particular semistandard lattices are supporting graphs. Our descriptions of these actions are explicit in the sense that relative to the bases obtained, the entries for the representing matrices of certain Lie algebra generators are rational coefficients we assign in pairs to the lattice edges. In Theorem 4.4 we show that if such coefficients can be assigned to the edges, then the assignment is unique up to products; we conclude that the associated weight bases enjoy certain uniqueness and extremal properties (the ``solitary'' and ``edge-minimal'' properties respectively). Our proof of this result is uniform and combinatorial in that it depends only on certain properties possessed by all semistandard posets. For certain families of semistandard lattices some of these results were obtained in previous papers; in Proposition 5.6 we explicitly construct new weight bases for a certain family of rank two symplectic representations. These results are used to help obtain in Theorem 5.1 the classification of those semistandard lattices which are supporting graphs.
Mathematics Subject Classification ID: 05E15
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 5132822
zbMATH Keywords: irreducible representations
zbMATH Keywords: Lie algebra generators
zbMATH Keywords: supporting graphs
zbMATH Keywords: weight bases
zbMATH Keywords: symplectic representations
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5132822