Vol. 258, No. 2, 2012

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Highest-weight vectors for the adjoint action of GLn on polynomials

Rudolf Tange

Vol. 258 (2012), No. 2, 497–510
Abstract

Let G = GLn be the general linear group over an algebraically closed field k, and let g = gln be its Lie algebra. Let U be the subgroup of G that consists of the upper unitriangular matrices. Let k[g] be the algebra of polynomial functions on g, and let k[g]G be the algebra of invariants under the conjugation action of G. For certain special weights, we give explicit bases for the k[g]G-module k[g]λU of highest-weight vectors of weight λ. For five of these special weights, we show that this basis is algebraically independent over k[g]G and generates the k[g]G-algebra r0k[g]U. Finally, we formulate the question whether in characteristic zero, k[g]G-module generators of k[g]λU can be obtained by applying one explicit highest-weight vector of weight λ in the tensor algebra T(g) to varying tuples of fundamental invariants.

Keywords
highest-weight vectors, semi-invariants, adjoint action
Mathematical Subject Classification 2010
Primary: 13A50
Secondary: 16W22, 20G05
Milestones
Received: 23 August 2011
Revised: 22 February 2012
Accepted: 27 February 2012
Published: 1 August 2012
Authors
Rudolf Tange
School of Mathematics
Trinity College Dublin
College Green
Dublin
Ireland