Timeline for Conjugacy for $p$-adic matrices of finite order
Current License: CC BY-SA 4.0
10 events
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| Jan 24, 2023 at 17:51 | history | edited | LSpice | CC BY-SA 4.0 |
Links, while this is on the front page
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| Aug 8, 2011 at 16:18 | comment | added | David E Speyer |
Note that this method also answers the question about $\mathbb{F}_p$ and $\mathbb{Q}_p$ conjugacy. There are only four indecomposable representations of $\mathbb{Q}_p[\mathbb{Z}/p^3]$, so the number of isomorphism classes of $\mathbb{Q}_p$ representations of dimension $n$ is $O(n^4)$ and the number of joint pairs $(\mathbb{F}_p \mathrm{-rep}, \mathbb{Q}_p \mathrm{-rep})$ is polynomial in $n$.
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| Aug 7, 2011 at 10:22 | comment | added | Tim Dokchitser | Very $ $nice! | |
| Aug 7, 2011 at 9:24 | vote | accept | Tim Dokchitser | ||
| Aug 6, 2011 at 13:27 | comment | added | Geoff Robinson | I am glad that is finally resolved. | |
| Aug 6, 2011 at 13:07 | comment | added | Frieder Ladisch | Really good. Dade shows more generally that $\mathbb{Z}_pG$ has indecomposable lattices of rank $kp^3$ whenever $\mathbb{Q}_pG$ has at least $4$ different irreducible representations. (In fact, the result is even more general.) Dade's paper is in the same issue of the Annals as the Heller-Reiner II, but is also in the first volume of Curtis-Reiner (Methods...). | |
| Aug 6, 2011 at 12:23 | comment | added | Alex B. | Thanks for the complement and for the reference, Kostia! | |
| Aug 6, 2011 at 12:18 | comment | added | user91132 | Very nice. The first paper by Heller and Reiner (Representations of Cyclic Groups in Rings of Integers, I) explains why there are only 3 indecomposable $\mathbf{Z}_p[C_p]$-modules in Theorem 2.6, and also treats the case $G = C_{p^2}$. The key idea is that any $\mathbb{Z}_p[C_p]$-module $M$ is an extension of a $B$-module $M/N$ by an $A$-module $N$; here $B = \mathbb{Z}_p[\zeta_p]$ and $A = \mathbb{Z}_p$ are both discrete valuation rings, so $M/N$ is a free $B$-module and $N$ is a free $A$-module whenever $M$ is $p$-torsion-free. Heller and Reiner explain how to handle the $Ext^1$ group. | |
| Aug 6, 2011 at 11:47 | history | edited | Alex B. | CC BY-SA 3.0 |
Added a remark about matrices of order p andd p^2
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| Aug 6, 2011 at 11:21 | history | answered | Alex B. | CC BY-SA 3.0 |