The existence of a polynomial factorization map for some compact linear groups

TL;DR

This paper proves that certain non-commutative, one-dimensional compact linear groups with specific representation properties admit polynomial factorization maps onto real vector spaces, expanding understanding of their algebraic structure.

Contribution

It establishes the existence of polynomial factorization maps for a class of non-commutative compact linear groups with particular representation characteristics.

Findings

Existence of polynomial factorization maps for specified groups.

Applicable to non-commutative, one-dimensional compact linear groups.

Groups have at least two faithful irreducible two-dimensional representations.

Abstract

It is proved that each of compact linear groups of one special type admits a polynomial factorization map onto a real vector space. More exactly, the group is supposed to be non-commutative one-dimensional and to have two connected components, and its representation should be the direct sum of three irreducible two-dimensional real representations at least two of them being faithful.