Image of Regular Unipotent under a Representation of ${\mathrm{G}\mathrm{L}}_3(\mathbb{C})$

We study the image of a regular unipotent element under any finite-dimensional irreducible polynomial representations of ${\mathrm{G}\mathrm{L}}_3(\mathbb{C})$. This problem is equivalent to decomposing certain compositions of irreducible representations as ${\mathrm{S}\mathrm{L}}_2(\mathbb{C})$-modules. We give an explicit decomposition of this finding, its Jordan decomposition.