Certain Linear Combinations of Exponential Functions are Positive under Semidefinite Linear Constraints

TL;DR

This paper introduces a group-theoretic approach to prove positivity of specific linear combinations of exponential functions under semidefinite linear constraints, leveraging the positivity of fusion coefficients for group characters.

Contribution

It presents a novel group-theoretic method to establish positivity of exponential function combinations under semidefinite constraints, connecting representation theory with functional inequalities.

Findings

Proves positivity of certain exponential combinations using group characters

Establishes a link between fusion coefficients and function positivity

Provides a new proof technique for semidefinite linear constraints

Abstract

In this article, I introduce a group-theoretical method to prove positivity of certain linear combinations (with coefficients generally lying in $\mathbb{C}$) of exponential functions under a set of semidefinite linear constraints. The basic group-theoretic fact we rely on is the positivity of the fusion coefficients for multiplication of group characters.