7
$\begingroup$

Let $p$ be a prime number and $n,l$ be natural numbers. I'm interested in the conditions under which the general linear groups of degree $n$ over the following two length $l$ finite discrete valuation rings with residue field $\mathbb{F}_p$ are isomorphic:

$$GL(n,\mathbb{Z}/p^l\mathbb{Z}) \cong GL(n,\mathbb{F}_p[t]/(t^l))$$

I've worked out the following cases:

  • The case $n = 1$ and $l = 2$: In this case, both groups are isomorphic to $\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/(p-1)\mathbb{Z}$.
  • The case $n = 1$ and $l > 2$: In this case, the only situation that the groups are isomorphic seems to be where $p = 2$ and ($l = 4$ or $l = 5$).
  • The case $n > 1$ and $l = 2$: The isomorphism seems to be equivalent to the requirement that the quotient map $GL(n,\mathbb{Z}/p^2\mathbb{Z}) \to GL(n,\mathbb{Z}/p\mathbb{Z})$ split. It seems that, for $n = 2$, this happens when $p = 2$ and when $p = 3$, but not when $p = 5$. This in turn seems to have something to do with modular/local representation theory. In particular, it seems that $GL(2,\mathbb{Z}/4\mathbb{Z}) \cong GL(2,\mathbb{F}_2[t]/(t^2))$ and $GL(2,\mathbb{Z}/9\mathbb{Z}) \cong GL(2,\mathbb{F}_3[t]/(t^2))$, but $GL(2,\mathbb{Z}/25\mathbb{Z}) \not \cong GL(2,\mathbb{F}_5[t]/(t^2))$. Is there a general way of figuring out what $p$ work and what don't?
  • The case $n > 1$ and $l > 2$: Don't know what happens here.

The question can be generalized somewhat to comparing the general linear group over a Galois ring with characteristic $p^l$ with residue field of size $q = p^r$ versus the ring $\mathbb{F}_q[t]/(t^l)$.

$\endgroup$
3
$\begingroup$

For $l=2$, the only cases where the extension splits are $n=2$ and $p=2,3$ and $n=3$ and $p=2$. In her PhD thesis, on page 3, Pooja Singla attributes this to [Chih Han Sah. Cohomology of split group extensions. II. J. Algebra, 45(1):17–68, 1977] and [Yuval Ginosar. On splitting of the canonical map: mod(p): GL(n,(p)) → GL(n,p). Comm. Algebra, 29(12):5879– 5881, 2001].

$\endgroup$
  • $\begingroup$ Thanks! I think that mostly settles the question (the l > 2 case is probably a finite amount of additional checking). $\endgroup$ – Vipul Naik Sep 12 '12 at 17:24

Your Answer

By clicking "Post Your Answer", you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.