Timeline for every element with eigenvalue 1
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| S Mar 12, 2015 at 1:24 | history | suggested | CommunityBot | CC BY-SA 3.0 |
Taking comments in consideration
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| Mar 12, 2015 at 0:47 | review | Suggested edits | |||
| S Mar 12, 2015 at 1:24 | |||||
| Mar 11, 2015 at 21:53 | comment | added | YCor | If $G$ is connected, if we consider an invariant flag $(V_i)$ of subspaces of $V=\mathbf{C}^n$ on which $G$ acts irreducibly on every $V_{i+1}/V_i$ then there exists $i$ such that every element of $G$ has eigenvalue 1 on $V_{i+1}/V_i$. So things boils down (for connected $G$) to the case when $G$ is reductive and acts irreducibly (take the Zariski closure if necessary), and actually semisimple (because the center acts by scalar matrices, hence is trivial). | |
| Mar 11, 2015 at 3:34 | comment | added | Noam D. Elkies | There are still counterexamples in ${\rm SO}_{2n}$. For example, a symmetric power of ${\rm SO}_m$ for $m$ odd (e.g. if $m \equiv 1 \bmod 4$ then ${\rm Sym}^2({\bf C}^m)$ is an even-dimensional orthogonal representation of ${\rm SO}_m$ each of whose elements has a $1$-eigenvector). | |
| Mar 11, 2015 at 2:54 | history | edited | curious | CC BY-SA 3.0 |
added 454 characters in body
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| Mar 11, 2015 at 2:16 | comment | added | Venkataramana | Every connected compact simple Lie group acts on its adjoint representation with the property that $1$ is an eigenvalue for every element. | |
| Mar 11, 2015 at 2:13 | comment | added | Noam D. Elkies | ${\rm SO}_n$, for $n>1$ odd? | |
| Mar 11, 2015 at 2:13 | comment | added | John Shareshian | Hmm, I did not see unipotent mentioned. Couldn't one take the complement to the trivial rep. in any linear rep. arising in the natural manner from a transitive subgroup of $S_n$ containing no $n$-cycle? | |
| Mar 11, 2015 at 1:57 | review | First posts | |||
| Mar 11, 2015 at 2:07 | |||||
| Mar 11, 2015 at 1:56 | history | asked | curious | CC BY-SA 3.0 |