Timeline for When is the intersection of cosets of a conjugacy class $0$-dimensional?
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| S Apr 30 at 9:05 | history | bounty ended | CommunityBot | ||
| S Apr 30 at 9:05 | history | notice removed | CommunityBot | ||
| S Apr 22 at 7:48 | history | bounty started | H A Helfgott | ||
| S Apr 22 at 7:48 | history | notice added | H A Helfgott | Canonical answer required | |
| Apr 18 at 9:40 | comment | added | H A Helfgott | I shoukd add that this question addresses a special case of mathoverflow.net/questions/469174/… | |
| Apr 18 at 7:26 | comment | added | H A Helfgott | Caveat: the same reasoning shows that there is a wider class of pairs $(h_1,h_2)$ for which the three varieties $\textrm{Cl}_g$, $h_1 \textrm{Cl}_g$, $h_2 \textrm{Cl}_g$ do not necessarily intersect transversally. For a lack of transversality, it is enough that $g^{-1}$, $I$, $h_1^{1}$ and $h_2^{-1}$ be linearly dependent. | |
| Apr 18 at 7:22 | comment | added | H A Helfgott | From that it's a short walk to conclusion (b) in the above. (I should really have said $\textrm{tr}(x) = c$, the set of $x$ satisfying $\textrm{tr}(x) = \textrm{tr}(h_1^{-1} x) = \textrm{\tr}(h^{-2} x)$, etc.) | |
| Apr 18 at 7:02 | comment | added | H A Helfgott | Let $n=2$. Then the conjugacy class of a regular semisimple element is given by the equation $\textrm{tr}(g) = c$, where $c\ne \pm 2$. The intersection above is thus the set of $g$ satisfying $\textrm{tr}(g) = \textrm{tr}(h_1^{-1} g) = \textrm{tr}(h_2^{-1} g) = c$. This is an intersection of three planes; the only way for it not to be a line or the empty set would be for $h_2^{-1}$ to equal $r h_1^{-1} + (1-r) I$ for some $r$, and it is easy to see that this cannot be the case for $h_1^{-1}$ semisimple and $h_2^{-1}$ distinct from $I$ and $h_1^{-1}$. | |
| Apr 18 at 7:01 | history | edited | H A Helfgott | CC BY-SA 4.0 |
added 96 characters in body
|
| Apr 18 at 6:47 | history | asked | H A Helfgott | CC BY-SA 4.0 |