Timeline for Peter-Weyl vs. Schur-Weyl theorem

7 events

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Apr 20, 2017 at 10:58	comment	added	Allen Knutson		For the tensor algebra, you need to combine different $a$. Then, you're asking for a statement on $\bigoplus_a \mathcal O(M_{a\times b})$, as in why should that thing have another, noncommutative, multiplication. What seems more reasonable is that it have a comultiplication, i.e. $\coprod_a M_{a\times b}$ has a monoid operation "concatenate vertically".
Apr 19, 2017 at 9:00	comment	added	Dan Petersen		I added a different answer to the question - if you have any input it would be appreciated!
Apr 19, 2017 at 5:28	comment	added	Dan Petersen		Thanks, this is really helpful! So what you're saying is that $(\mathbb C^b)^{\otimes a} = \bigoplus_{\lambda \vdash a} \sigma_\lambda \otimes (\mathbb C^b)_\lambda$ sits inside of $\mathcal O(M_{a \times b}) = \bigoplus_{\lambda} (\mathbb C^a)_\lambda \otimes (\mathbb C^b)^\ast_\lambda$ in a nice and conceptual way that respects the decompositions. Is there a statement that also relates the multiplication in the tensor algebra and the multiplication in the coordinate ring?
Apr 19, 2017 at 5:19	history	edited	Dan Petersen	CC BY-SA 3.0	edited body
Apr 19, 2017 at 4:35	history	edited	Allen Knutson	CC BY-SA 3.0	added 156 characters in body
Apr 19, 2017 at 4:13	history	edited	Allen Knutson	CC BY-SA 3.0	added 999 characters in body
Apr 19, 2017 at 0:25	history	answered	Allen Knutson	CC BY-SA 3.0