Questions tagged [schur-functors]
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7
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How to constructively/combinatorially prove Schur-Weyl duality?
How is Schur-Weyl duality (specifically, the fact that the actions of the group ring $\mathbb{K}\left[ S_{n}\right] $ and the monoid ring
$\mathbb{K}\left[ \left( \operatorname*{End}V,\cdot\...
24
votes
2
answers
2k
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Direct proof that the centralizer of $GL(V)$ acting on $V^{\otimes n}$ is spanned by $S_n$
Let $V$ be a finite dimensional vector space over a field of characteristic zero. Let $A$ be the space of maps in $\mathrm{End}(V^{\otimes n})$ which commute with the natural $GL(V)$ action. Clearly, ...
7
votes
1
answer
515
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Question about decomposition of exterior product
In their paper "New lower bounds for the border rank of matrix multiplication", Landsberg and Ottaviani make use of the fact that
$$\tag{$\dagger$} {\textstyle\bigwedge}^p(V\otimes W) \cong \...
6
votes
1
answer
406
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Yoneda map for a composition of a representable functor and an arbitrary functor
Let $\mathcal{C}$ and $\mathcal{D}$ be categories and let $T : \mathcal{C} \rightarrow \mathcal{D}$ be a functor. Suppose that $F : \mathcal{D}^\mathrm{op} \rightarrow \mathrm{Set}$ is a functor. (So ...
5
votes
3
answers
609
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An isomorphism of 2-Schur modules
This is the little brother of question 68071: elementary, simple-looking and probably much easier to answer. Of course, it is just a small part of question 68071, as anybody with $\lambda$-rings ...
4
votes
1
answer
213
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A question on complex semisimple Lie groups and $(\mathbb{C}^2, \omega)$
Consider $(\mathbb{C}^2, \omega)$ where $\omega$ is a non-degenerate complex skew-symmetric bilinear form on $\mathbb{C}^2$. Let us write
$V = (\mathbb{C}^2, \omega)$
There are many spaces one can ...
4
votes
1
answer
612
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Can the projection (tensor algebra) -> (symmetric algebra) be forced to split in char. p by factoring out p-th powers?
Question 1 (the weak and simple statement, which, I think, already is wrong): Let $p$ be a prime. Let $k$ be a field with characteristic $p$.
For any $k$-vector space $V$, consider the canonical ...