Orthogonal bases of Brauer relative symmetric polynomials for certain groups

TL;DR

This paper investigates orthogonal bases of symmetry classes of polynomials linked to Brauer characters of Semi-Dihedral and Dihedral groups, providing conditions for their existence and exploring tensor symmetrization.

Contribution

It introduces new criteria for the existence of orthogonal bases of symmetry classes of polynomials associated with Brauer characters of specific groups.

Findings

Established necessary and sufficient conditions for orthogonal bases.

Analyzed symmetry classes of polynomials for Semi-Dihedral and Dihedral groups.

Provided insights into tensor symmetrization using Brauer characters.

Abstract

In this paper, we discuss O-basis of symmetry classes of polynomials associated with the Brauer character of the Semi-Dihedral groups and Dihedral groups. Also, necessary and sufficient conditions are given for the existence of an orthogonal basis consisting of standard (decomposable) symmetrized tensors for the class of tensors symmetrized using a Brauer character of the Semi-Dihedral groups.