Kernel solutions of the Kostant operator on eight-dimensional quotient spaces

TL;DR

This paper derives explicit kernel solutions of the Kostant operators on specific eight-dimensional quotient spaces related to Lie algebras, using Schwinger's oscillators to express generators and representations.

Contribution

It provides explicit solutions for the Kostant operators on ${ m su}(5)/{ m su}(4) imes { m u}(1)$ and ${ m so}(6)/{ m so}(4) imes { m so}(2)$ quotient spaces, expanding understanding of these operators.

Findings

Derived kernel solutions for ${ m su}(5)/{ m su}(4) imes { m u}(1)$

Derived kernel solutions for ${ m so}(6)/{ m so}(4) imes { m so}(2)$

Expressed solutions in terms of diagonal subalgebras

Abstract

After introducing the generators and irreducible representations of the ${\rm su}(5)$ and ${\rm so}(6)$ Lie algebras in terms of the Schwinger's scillators, the general kernel solutions of the Kostant operators on eight-dimensional quotient spaces ${\rm su}(5)/{\rm su}(4)\times {\rm u}(1)$ and ${\rm so}(6)/{\rm so}(4)\times {\rm so}(2)$ are derived in terms of the diagonal subalgebras ${\rm su}(4)\times {\rm u}(1)$ and ${\rm so}(4)\times {\rm so}(2)$, respectively.