# Low dimensional representations of $SL_n(\mathbb{Z}/p^\ell \mathbb{Z})$

When $$\ell = 1$$ I know that the smallest non-trivial irreducible complex representations of $$SL_n(\mathbb{Z}/p\mathbb{Z})$$ has dimension $$\frac{p^n - 1}{p-1} - 1$$ (with maybe some exceptions for small values of $$n$$ or $$p$$).

For $$\ell > 1$$ we have the map $$SL_n(\mathbb{Z}/p^\ell \mathbb{Z}) \to SL_n(\mathbb{Z}/p\mathbb{Z})$$ so we still get a representation of this size. I was wondering if this is still the smallest possible dimension (again maybe with a finite number of exceptions for small $$p$$ and $$n$$), and if not are there known lower bounds for how small such a representation can be?

I found some calculations of characters for small values of $$n$$ that suggest this could be true, but otherwise couldn't find much. Is there anywhere in the literature that addresses this sort of question? I'm also interested in similar results for $$Sp_{2n}(\mathbb{Z}/p^\ell \mathbb{Z})$$.

• Look at this paper, (Theorem 1): arxiv.org/pdf/1202.4194.pdf – MO B Oct 24 '18 at 2:15
• Maybe you mean "some exceptions for small $n$ OR $p$"? (The minimal dimension for $n=2$ is $(p-1)/2$ for any large $p$.) – user148212 Oct 24 '18 at 5:48
• @user148212 Ah good catch, the context where this came up for me has $n>2$, but you are absolutely correct. – Nate Oct 24 '18 at 20:18

For $$n=2$$ and $$p$$ odd:
The minimal possible dimension of a non-trivial irrep of $$G_l=\mathrm{SL}_2(\mathbb{Z}/p^l)$$ is always $$(p-1)/2$$, and the minimal possible dimension of a non-trivial primitive irrep of $$G_l$$ is $$p^{l-2}(p^2-1)/2$$.