Let $S^kV$ be the $k$-th symmetric power of tautological representation of $U(n)$ how to see that it's irreducible? I'm trying to do it using weight, but with no benefits..
2026-04-28 19:04:29.1777403069
Symmetric power of tautological representation of $U(n)$
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A direct argument is possible, but it's a bit messy. It's nicer to work at the Lie algebra level because there it's easier to write down maps that move between weight spaces. An even nicer argument uses Schur-Weyl duality, which also implies that the exterior powers $\wedge^k V$ are irreducible and a bunch of other nice things.