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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