Timeline for Fixed subspaces of a family of representations $\rho_t: F_2\to GL(n,\mathbb C)$
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 23, 2021 at 22:18 | comment | added | aglearner | Mikael, thanks! I got it. So the matrix $(A-1)^*(A-1)$ is non-negative definite, and it vanishes exactly on the kernel of $A-1$. Same for $(B-1)^*(B-1)$. Then indeed, if this equality holds for an interval, it holds for the whole $\mathbb R$. That's nice. What is also great is that this generalises to representations of any finitely generated group. | |
| Jun 23, 2021 at 21:13 | answer | added | Chris | timeline score: 5 | |
| Jun 23, 2021 at 19:06 | comment | added | Mikael de la Salle | Hint: two matrices $A$ and $B$ have a common eigenvector with eigenvalue $1$ if and only if $\mathrm{det}((A-1)^* (A-1)+(B-1)^* (B-1))=0$. | |
| Jun 23, 2021 at 17:59 | comment | added | aglearner | Chris, thanks, sorry, I fixed the misprint | |
| Jun 23, 2021 at 17:58 | history | edited | aglearner | CC BY-SA 4.0 |
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| Jun 23, 2021 at 17:56 | comment | added | Chris | You write $A_t(v_t)=B_t(v_t)=0$ for $v_t\neq 0$ but $A_t, B_t$ are invertible, thus their kernel is zero. | |
| Jun 23, 2021 at 17:32 | history | asked | aglearner | CC BY-SA 4.0 |