# Immanant inequalities and weight spaces

## TL;DR

This paper develops a trace formula for immanants of submatrices using weight spaces, extends Kostant's formula, and establishes new inequalities and criteria for non-vanishing immanants in positive matrices.

## Contribution

It introduces a generalized trace formula for immanants based on weight spaces, extending Kostant's work, and provides new non-vanishing criteria and inequalities for positive matrices.

## Key findings

- Derived a trace formula for immanants on generalized principal submatrices.
- Established a criterion for non-vanishing immanants in positive definite matrices.
- Presented a unifying inequality encompassing Kostant, Schur, and Stembridge's inequalities.

## Abstract

We first obtain a trace formula for immanants of generalized principal submatrix of any complex matrix based on any weight space for finite dimensional representations of the general linear group. Our trace formula contains Kostant's famous formula for immanants on $0$-weight spaces as special case. We then present a criterion for non-vanishing immanants for any generalized principal submatrix of positive definite Hermitian or nonsingular totally nonnegative matrices, which strengthened the well-known results of Schur and Stembridge. Furthermore, we present an inequality that contains Kostant, Schur and Stembridge's famous inequalities as special cases.