Semidefinite programming bounds for complex spherical codes

TL;DR

This paper develops semidefinite programming bounds for complex spherical codes by analyzing the irreducible decomposition under the unitary group's action, aiding in establishing upper bounds for codes with specific inner products.

Contribution

It introduces a method to determine semidefinite programming bounds for complex spherical codes using irreducible decomposition under the unitary group action.

Findings

Derived the irreducible decomposition under the unitary group action.

Established semidefinite programming bounds for complex spherical codes.

Provided a framework for upper bounds with prescribed inner products.

Abstract

A complex spherical code is a finite subset on the unit sphere in $\mathbb{C}^d$. A fundamental problem on complex spherical codes is to find upper bounds for those with prescribed inner products. In this paper, we determine the irreducible decomposition under the action of the one-point stabilizer of the unitary group $U(d)$ on the polynomial ring $\mathbb{C}[z_1\ldots,z_d,\bar{z}_1,\ldots,\bar{z}_d]$ in order to obtain the semidefinite programming bounds for complex spherical codes.