Harada's conjecture II and Gramian determinants

TL;DR

This paper investigates Harada's conjecture II by examining Harada's number through Gramian determinants, introduces related invariants, and provides conditions and examples to verify the conjecture.

Contribution

It introduces a new approach using Gramian determinants to analyze Harada's conjecture II and defines invariants related to central characters.

Findings

Provides a necessary and sufficient condition for Harada's conjecture II to hold.

Calculates Harada's numbers explicitly in specific examples.

Introduces invariants generalizing the square of Harada's number.

Abstract

Let $G$ be a finite group. Harada's conjecture II states that the ratio of the product of all the number of elements in conjugacy classes over that of all degrees of irreducible complex characters of $G$ is an integer. The ratio is called Harada's number. In this article, we discuss the Harada's number from the view point of Gramian determinants for suitable inner spaces and introduce an invariants generalizing the square of Harada's number associated to central characters of $G$. We also give a necessary and sufficient condition so that Harada's conjecture II holds. We calculate explicitly Harada's numbers in some examples by using the method given in this article.