Field of Borel Group Invariant of adjoint representatnion of the group GL(n, K)

TL;DR

This paper explicitly constructs generators for the invariant fields of the Borel and unitriangular groups acting on the adjoint representation of GL(n,K), advancing invariant theory.

Contribution

It provides explicit generators for the invariant fields of the Borel and unitriangular groups in the adjoint representation of GL(n,K), including proofs of their algebraic independence.

Findings

Generators for the invariant field of the unitriangular group are explicitly given.

Generators for the Borel group invariant field are constructed and proven algebraically independent.

The work advances explicit understanding of invariant fields in Lie group representations.

Abstract

The paper is devoted to invariant theory problems. In particular, to the problem of finding generators of invariant fields in an explicit form. The set of generators is given for invariant field of unitriangular group of adjoint representation of $GL(n,K)$. Moreover, the set of generators of borel group invariant field is constructed and their algebraic independence is proved. Keywords: Lie group, adjoint representation, the field of invariant, generators of the field of invariant, Borel group.