Conformal Galilei algebras, symmetric polynomials and singular vectors

TL;DR

This paper classifies homomorphisms of Verma modules for conformal Galilei algebras in one dimension, linking singular vectors to symmetric polynomials and providing explicit descriptions of these mappings.

Contribution

It provides a complete classification and explicit description of homomorphisms of Verma modules for conformal Galilei algebras, connecting singular vectors with symmetric polynomial expansions.

Findings

Homomorphisms are determined by singular vectors solving differential operators.

Singular vectors correspond to coefficients in symmetric polynomial expansions.

Explicit formulas for homomorphisms are derived for all integer ll.

Abstract

We classify and explicitly describe homomorphisms of Verma modules for conformal Galilei algebras $\mathfrak{cga}_\ell(d,{\mathbb C})$ with $d=1$ for any integer value $\ell \in \mathbb{N}$. The homomorphisms are uniquely determined by singular vectors as solutions of certain differential operators of flag type, and identified with specific polynomials arising as coefficients in the expansion of a parametric family of symmetric polynomials into power sum symmetric polynomials.