Timeline for How to constructively/combinatorially prove Schur-Weyl duality?
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| Sep 17, 2020 at 16:25 | comment | added | Claudio Procesi | You should also look at the recent book with De Concini on matrices. There we develop one more time the straightening laws in order to prove the first and second fundamental theorem of invariants of matrices in a characteristic free way without using algebraic geometry.I think we explain combinatorially sevral points of interest. | |
| Aug 30, 2020 at 16:30 | comment | added | darij grinberg | That said, a shorter or simpler proof would still be appreciated. Even though all of what you cite from the Doubilet/Rota/Stein paper [2] is nowadays available (somewhat implicitly) in the 7-page expository note Richard G. Swan, On the straightening law for minors of a matrix, arXiv:1605.06696v1, the proof still amounts to something like 15 pages. (Impressive work, though!) | |
| Aug 30, 2020 at 16:27 | review | Late answers | |||
| Aug 30, 2020 at 16:41 | |||||
| Aug 30, 2020 at 16:20 | comment | added | darij grinberg | ... the rational function $f\left(\alpha x_1, \alpha x_2, \ldots, \alpha x_m, \alpha \xi_1, \alpha \xi_2, \ldots, \alpha \xi_m\right)$. But the latter function belongs to $B\left[1/d\right]$ since all coordinates of each of the vectors $\alpha x_1, \alpha x_2, \ldots, \alpha x_m$ and each of the covectors $\alpha \xi_1, \alpha \xi_2, \ldots, \alpha \xi_m$ do. (This is what you check in the computational proof of Lemma 3.2.) | |
| Aug 30, 2020 at 16:19 | comment | added | darij grinberg | Thank you -- I actually realized the same a few weeks ago, after Henning Krause reminded me of your paper. Theorem 4.1 is Schur-Weyl duality. The geometric language in the proof of Lemma 3.2 used to scare me off, but I realized none of it is actually being used: Just take the $n\times n$-matrix whose columns are $x_1, x_2, \ldots, x_n$, and let $\alpha \in \operatorname{GL}_n\left(S\left[1/b\right]\right)$ be its inverse; then, any invariant in $f \in S\left[1/d\right]^G$ is invariant under the action of $\alpha$, and thus can be rewritten as a power of $d$ times ... | |
| Aug 30, 2020 at 16:13 | comment | added | მამუკა ჯიბლაძე | Welcome to Mathoverflow, professor Procesi! | |
| Aug 30, 2020 at 16:10 | review | First posts | |||
| Aug 30, 2020 at 17:44 | |||||
| Aug 30, 2020 at 16:07 | history | answered | Claudio Procesi | CC BY-SA 4.0 |