19
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How to constructively/combinatorially prove Schur-Weyl duality?
This is a quick answer to explain the statement that the hard direction of Schur-Weyl duality is the same thing the First Fundamental Theorem of invariant theory.
Let $V$ be a finite dimensional ...
- 151k
10
votes
Accepted
Coefficients when rewriting the Hook-Content polynomials in terms of binomial polynomials
These coefficients can be understood using the theory of $(P,\omega)$-partitions, as discussed for example in Section 3.15 of Stanley's "Enumerative Combinatorics," Volume 1, 2nd edition.
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8
votes
Accepted
Yoneda map for a composition of a representable functor and an arbitrary functor
This property of the functor $T$ is called being a dense functor, or "densely generating functor". The notion was first introduced by Isbell in the case where $T$ is fully faithful under the ...
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8
votes
Accepted
Details about plethysm
Of the three plethysms in 3), $h_n[h_2]$ is the simplest. A "modern" proof can be found for instance in Example A2.9 (page 449) of Enumerative Combinatorics, vol. 2. This was known to D. E. Littlewood ...
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8
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How to constructively/combinatorially prove Schur-Weyl duality?
look at
C. De Concini, C. Procesi, A characteristic free approach to invariant theory, Adv. Math. 21 (1976),
330–354.
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8
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How to constructively/combinatorially prove Schur-Weyl duality?
This a continuation of my first answer. I was trying to edit the previous one but the MathJax processing was freezing my computer. I suppose that answer was getting too long.
@Darij: There has been ...
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7
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How to constructively/combinatorially prove Schur-Weyl duality?
Since you are OK with an explicit combinatorial proof in characteristic zero I think you should also be OK with working
over $\mathbb{C}$. Part b) of Schur-Weyl duality follows from the First ...
- 21.8k
6
votes
Accepted
A question on complex semisimple Lie groups and $(\mathbb{C}^2, \omega)$
I think that the kind of question you are asking is one that was treated by Dynkin back in the 1950s (see Semisimple subalgebras of semisimple Lie algebras. (Russian) Mat. Sbornik N.S. 30(72), (1952), ...
- 106k
2
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Young Symmetrizer and Exterior Products, such as $S_{(2,1)}V = Ker(\Lambda^2V \otimes V \to \Lambda^3V )$
Question 1: The general construction of the Schur module (see for instance Fulton's book on Young tableaux, Chapter 8; the example $S_{2,1}(V)$ appears in the introduction of Part II) yields
$S_{2,1}(...
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