I am not very familiar with representation theory, but I keep seeing it applied in very different contexts. I read applications of representation theory for decompositions of space of polynomials, like the harmonic decomposition or, more generally, the Fischer decomposition. Using abstact results from Lie groups and Lie algebras, like commutation relations and Howe duality, it is possible to obtain very concrete results, for example the harmonic decomposition
$$
\mathcal{P}_k=\bigoplus_{j=0}^{\lfloor k/2\rfloor}r^{2j}\mathcal{H}_{k-2j}.
$$
I would like to understand which paths follow the proofs of these results and how to study the Lie algebras to get similar decompositions.
Thank you in advance
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Jules Binet
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