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 Sep 24 awarded Autobiographer Jun 11 revised Previous derivation of a combinatorial formula for the fundamental representations of $A_{n-1}$ deleted 7 characters in body Jun 11 asked Previous derivation of a combinatorial formula for the fundamental representations of $A_{n-1}$ Jun 7 comment Symmetric powers of Schur polynomials Many thanks Steven. I will look into this interface Jun 7 accepted Symmetric powers of Schur polynomials May 31 comment Symmetric powers of Schur polynomials Per Alexandersson: I do not believe that there is such a rule for composition of Schur functions. There exists the Littlewood-Richardson rule for multiplication of Schur functions that can be phrased as you say, but this is surely known to you. May 29 comment Symmetric powers of Schur polynomials Let's consider the representation of $A_2$ i.e., $\mathfrak{sl}_3$ given by $\lambda = (3,2)$ so this has highest weight $\omega_1 + 2\omega_2$ where $\omega_i$ is the $i$th fundamental weight of $A_2$, and I write an arbitrary irreducible representation of $A_2$ as $\Gamma_{a,b}$ where $\lambda = a\omega_1 + b\omega_2$ is the highest weight. Then $Sym^2$ applied to this Schur polynomial $s_{(3,2)}$ should yield the Schur polynomials $s_{(2,2)} + s_{(3,1)} + s_{(4,3)} + s_{(4,0)} + s_{(6,4)}$. Is this what you meant? May 29 asked Symmetric powers of Schur polynomials May 24 awarded Supporter May 24 accepted Decomposition into irreducibles of symmetric powers of irreps. May 24 comment Decomposition into irreducibles of symmetric powers of irreps. This is quite helpful as it gives me a new direction in which to read. I will look through Goodman-Wallach and continue with my study of Fulton and Harris. Thank you for your response Dr. Humphreys May 23 comment Decomposition into irreducibles of symmetric powers of irreps. Hmmm, I am looking through his publications here: manchester.ac.uk/research/Roger.bryant/publications and he seems to treat with Lie algebras in characteristic $p$, but I should have mentioned that I am working over $\Bbb C$ here. And the symmetric powers he deals with are mostly with finite groups? Was there a particular paper you had in mind? Many thanks for your time. May 23 asked Decomposition into irreducibles of symmetric powers of irreps. May 18 comment How to detect if a subgroup lands inside an orthogonal group? @Aakumadula thank you. I will consult Fulton and Harris, I will surely find it in there May 18 asked How to detect if a subgroup lands inside an orthogonal group? May 14 awarded Scholar May 14 accepted Finding spherical representations of $GL(n, \mathbb{C})$. May 14 comment Finding spherical representations of $GL(n, \mathbb{C})$. Thanks for the information Marc, this is helpful May 14 comment Finding spherical representations of $GL(n, \mathbb{C})$. @David that article is indeed interesting; however, I believe that my further edits will help to clarify exactly what I seek. Thank you for finding this though, and for those who do not have MathSciNet I have found a freely available version of the article here: eudml.org/doc/89398 in case a viewer of this question is indeed looking for the above classification. May 14 revised Finding spherical representations of $GL(n, \mathbb{C})$. clarified the objective of the question