# Some remarks on permanent dominant conjecture

## TL;DR

This paper establishes a new identity linking determinants and generalized matrix functions, and offers a criterion for positive semi-definite matrices that supports the permanent dominant conjecture, generating numerous classes of such matrices.

## Contribution

It introduces a novel identity and a new criterion for positive semi-definite matrices related to the permanent dominant conjecture, expanding the set of known applicable matrices.

## Key findings

- Derived an identity between determinant and generalized matrix function.
- Provided a criterion affirming the permanent dominant conjecture.
- Generated infinitely many classes of positive semi-definite matrices supporting the conjecture.

## Abstract

In this paper we provide an identity between determinant and generalized matrix function. Also, a criterion of positive semi-definite matrices affirming the permanent dominant conjecture is given. As a consequence, infinitely many infinite classes of positive semi-definite serving the conjecture (does not depend on groups or characters) are provided by generating from any positive semi-definite matrix having no zero in the first column.