Polynomial ring representations of endomorphisms of exterior powers

An explicit description of the ring of the rational polynomials in r indeterminates as a representation of the Lie algebra of the endomorphisms of the k-th exterior power of a countably infinite-dimensional vector space is given. Our description is based on results by Laksov and Throup concerning the symmetric structure of the exterior power of a polynomial ring. Our results are based on approximate versions of the vertex operators occurring in the celebrated bosonic vertex representation, due to Date, Jimbo, Kashiwara and Miwa, of the Lie algebra of all matrices of infinite size, whose entries are all zero but finitely many.

Exterior Powers and Pointwise Creation Operators

We develop a theory of pointwise wedge products of vector-valued functions on the circle and the disc, and obtain results which give rise to a new approach to the analysis of the matricial Nehari problem. We investigate properties of pointwise creation operators and pointwise orthogonal complements in the context of operator theory and the study of vector-valued analytic functions on the unit disc.

Exterior powers of the standard E6-module: An elementary approach

For the standard [Formula: see text]-dimensional representation [Formula: see text] of the exceptional group [Formula: see text] of type [Formula: see text] we prove that [Formula: see text] is a Donkin pair if and only if the characteristic of a ground field is greater than [Formula: see text]. We also develop an elementary approach to describe submodule structure of any exterior power of [Formula: see text].