Timeline for Exponentiation of vector spaces?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Jun 27, 2023 at 17:19 | comment | added | LSpice | This construction still makes perfectly good sense over any ring, in which generality it (more or less formally) is monoidal in $W$ (converting $\oplus$ canonically to $\otimes$); but I can't seem to show that it is monoidal in $V$ (converting $\otimes$ canonically to $\otimes$) unless $W$ is free of finite rank—which is, of course, no problem in the generality in which the problem was proposed, but makes me wonder whether it's not true in general, or I'm just not clever enough. | |
| Jun 24, 2023 at 19:00 | comment | added | LSpice | @AllenKnutson, re, doesn't the argument about different $\operatorname{GL}(W)$ representations already rule that out? For example, if $T - I$ is $0$ on $V$ and maps $1$ non-$0$-ly into $V$, then $T^{\otimes W} - I$ has nowhere to $\operatorname{GL}(W)$-equivariantly map the $1$ summand; whereas, if $T - I$ is $0$ on $1$ and maps $V$ non-$0$-ly to $1$, then $T^{\otimes W} - I$ has nowhere to map the $\bigwedge^\text{top}W \otimes V^{\otimes\text{top}}$ summand, at least if $\dim(\text{top}) \ne 0$ as a scalar. | |
| Feb 10, 2018 at 5:43 | vote | accept | მამუკა ჯიბლაძე | ||
| Oct 1, 2015 at 0:06 | comment | added | Allen Knutson | So this is natural w.r.t. the $GL(V)$ action. How about either of the parabolics containing it? (I.e. replace $1\oplus V$ by an extension of $V$ by $1$, or of $1$ by $V$). | |
| Sep 30, 2015 at 6:57 | comment | added | მამუკა ჯიბლაძე | ...no, this analogy is not quite correct, as $V$ takes place of $k$ here. Still there is some connection I think | |
| Sep 30, 2015 at 6:50 | comment | added | მამუკა ჯიბლაძე | This is somehow reminiscent of the chain complex structure (I think due to Illusie?) on $\Lambda^*(W)\otimes\operatorname{S}^*(W)$ quasiisomorphic to $1$, with (super-) $\Lambda^*(W)$ being "like $(1-k)^{\otimes W}$" and $\operatorname{S}^*(W)$ "like $(1-k)^{\otimes(-W)}$" - yours looks like a "noncommutative version" of it with tensor algebra in place of the symmetric algebra... | |
| Sep 30, 2015 at 5:35 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
added 321 characters in body
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| Sep 30, 2015 at 5:16 | history | answered | Qiaochu Yuan | CC BY-SA 3.0 |