Timeline for Vector spaces with natural bases
Current License: CC BY-SA 2.5
5 events
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| Jul 19, 2010 at 7:58 | comment | added | Victor Protsak | In the case of modular forms, the multiplicity one property is really deep. There are more elementary examples of a full eigenvector basis of a max commuting algebra where everything works, but the signed basis issue is typically unavoidable. | |
| Jul 19, 2010 at 7:55 | comment | added | Victor Protsak | Qiaochu, the key property you need is "multiplicity one", but note that, in general, a decomposition into 1-dim subspaces doesn't yield a basis, not even a "signed basis". In the case of the root space decomposition of a semisimple Lie algebra $\mathfrak{g},$ it holds for the orthogonal complement of the Cartan subalgebra $\mathfrak{h}\subset \mathfrak{g}$, but not for $\mathfrak{h}$ itself, so you don't even get a decomposition into 1-dim subspaces, and there are at least 2 choices of "normalization where the structure constants are as nice as possible" differing by signs. | |
| Jul 19, 2010 at 1:05 | history | made wiki | Post Made Community Wiki by S. Carnahan♦ | ||
| Jul 19, 2010 at 1:00 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
added 340 characters in body
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| Jul 18, 2010 at 23:43 | history | answered | Qiaochu Yuan | CC BY-SA 2.5 |