| Summary: | Motivated by describing the symmetry of a theoretical model of dressed photons, we introduce several spaces with Lie group actions and the morphisms between them depending on three integer parameters <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mi>r</mi><mo>≥</mo><mi>s</mi></mrow></semantics></math></inline-formula> on dimensions. We discuss the symmetry on these spaces using classical invariant theory, orbit decomposition of prehomogeneous vector spaces, and compact reductive homogeneous space such as Grassmann manifold and flag variety. Finally, we go back to the original dressed photon with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>=</mo><mn>4</mn><mo>,</mo><mi>r</mi><mo>=</mo><mn>2</mn><mo>,</mo><mi>s</mi><mo>=</mo><mn>1</mn></mrow></semantics></math></inline-formula>.
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