Timeline for Orbit of the identity matrix under Lie group algebra actions
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
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| Mar 26, 2012 at 1:40 | comment | added | John Jiang | @Jim: No problem. I am glad ppl here helped me understand the actions on matrices, which is crucial for my study of the Kac random walk. See for instance: arxiv.org/abs/0905.1539 | |
| Mar 25, 2012 at 22:15 | comment | added | Jim Humphreys | @John: Sorry, I didn't understand what you meant by "the action" here, since I took it to be just left multiplication in the matrix algebra. | |
| Mar 25, 2012 at 20:59 | vote | accept | John Jiang | ||
| Mar 25, 2012 at 20:06 | answer | added | Claudio Gorodski | timeline score: 3 | |
| Mar 25, 2012 at 20:01 | comment | added | John Jiang | @Claudio: thanks! That's pretty much as far as I could get, and for $n=2$ that's handy. | |
| Mar 25, 2012 at 19:59 | comment | added | John Jiang | @Jim: I am actually referring to the action on the space of $n\times n$ matrices. So for instance when n=2, the vector space spanning by $\mathbb{R}SO(2)$ acting on $I_2$ is simply $\mathbb{R}I_2 \oplus \mathbb{R} J_2$, which is not the full $M_{2 \times 2}$. Hope this clarifies a bit. I could be hallucinated.. | |
| Mar 25, 2012 at 19:59 | comment | added | Claudio Gorodski | One simple remark is that the vector space spanned by the orbit $Gp$ always contains the vector space $\mathfrak g\cdot p$, where the Lie algebra $\mathfrak g$ is acting by the derived representation. | |
| Mar 25, 2012 at 19:57 | comment | added | Claudio Gorodski | It is the sum of $n$ copies of the natural actio n, so it is not irreducible. | |
| Mar 25, 2012 at 19:53 | comment | added | Jim Humphreys | @John: Unless I'm missing something subtle, the "natural" group action on the underlying vector space is irreducible and thus the resulting matrices should span the whole space. Is there more going on? | |
| Mar 25, 2012 at 19:49 | history | edited | John Jiang | CC BY-SA 3.0 |
added 217 characters in body; added 174 characters in body
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| Mar 25, 2012 at 19:47 | comment | added | John Jiang | @Jim: yes just the usual left multiplication action. So the question is really asking for the vector space spanned by $\sum_i c_i A_i$, where $A_i \in SO(n)$. I will clarify in the text. | |
| Mar 25, 2012 at 19:46 | comment | added | Jim Humphreys | The question isn't clear to me. What is the precise action here, and are you just referring to the group algebra of the abstract group? | |
| Mar 25, 2012 at 19:40 | history | edited | John Jiang | CC BY-SA 3.0 |
deleted 200 characters in body; edited title
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| Mar 25, 2012 at 19:33 | history | asked | John Jiang | CC BY-SA 3.0 |