the character tables of irreducible representations of $SL(3,Z_q)$

Question

The following paper gives a classification of the character tables of irreducible representations of $SL(3,GF(q))$ where $q$ is a power of a prime number, and $ GF(q)$ a finite field of $q$ elements.

WILLIAM A. SIMPSON AND J. SUTHERLAND FRAME Can. J. Math., Vol. XXV, No. 3,1973, pp. 486-494 THE CHARACTER TABLES FOR SL(3, q ), SU(3, *•), PSL(3, q), PSU(3, q *)

Here I would to ask "do we have a classification of the character tables of irreducible representations of $SL(3,Z_q)$, where $Z_q=Z/qZ$?"

Answer 1

As was already mentioned, the answer to the question asked is "no, we currently do not have a good classification". Here I wish to describe a successful and interesting recent line of research which does not aim at giving such a classification, rather merely at counting how many representations we do have.

To put things in context, let me first make the following observations:

• the set of irreducible representations of a product of groups is in bijection with the product of the sets of irreducible representations of the groups.

• $\text{SL}_n$ over a product of rings (commutative with 1) is isomorphic to the product of $\text{SL}_n$ over the rings.

• for a natural $m$, $\mathbb{Z}/m\mathbb{Z}$ is the product of the rings $\mathbb{Z}/q\mathbb{Z}$ where $q$ is a prime power.

Thus, the study of the rep theory of $\text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ is naturally reduced to the study of the rep theories of the groups $\text{SL}_n(\mathbb{Z}/p^k\mathbb{Z})$ for prime $p$. Fixing $p$ and varying $k$, these are grouped together as the (continuous) representation theory of $\text{SL}_n(\mathbb{Z}_p)$, where $\mathbb{Z}_p$ is the ring of $p$-adic integers.

For $n=2$ the irreducible representations of $\text{SL}_n(\mathbb{Z}_p)$ are indeed classified, as appeared in the answere by Jim Humphreys. To this answer I wish to add the reference to Uri Onn's paper in which he classifies all irreducible representations over an arbitrary discrete valuation ring. See also a comment below by A Stasinski.

For $n\geq 3$ the representation theory of $\text{SL}_n(\mathbb{Z}_p)$ is much more complicated and a classification is currently out of reach. However we do know that $\text{SL}_n(\mathbb{Z}_p)$ has only finitely many irreducible representations at each dimension $d$. Denoting this number by $r_d$ we are interested in the study of the sequence $(r_d)_{d=1}^\infty$. On this problem there have been in recent years a remarkable progress on which I wish to report, and this is why I am writing this answer.

For $n=3$ it is shown in by Avni-Klopsch-Onn-Voll that the sequence $r_d$ grows similarly to $d^{3/2}$. For a precise statement see their Duke paper. Recently Aizenbud-Avni got a uniform statement for all $n$, that is $r_d$ is $O(d^{22})$. For this see their Inventiones paper. Whether such a uniform bound exists was asked as a question in Larsen-Lubotzky.

I would be happy to tell more about the techniques of these remarkable papers, but this answer is already getting too long. You will have to read through the sources. Let me just mention that in order to study the sequence $(r_d)$ we set the representation zeta function $\zeta(s)=\sum r_d\cdot d^{-s}$. A general result of Jaikin-Zapirain tells us that this function is rational in the variable $p^{-s}$. The representation growth is then related to the study of the poles of this zeta function. Let me also mention that there is a global theory here, which comes from the representation theory of $\text{SL}_n(\mathbb{Z})$.

Edit: As remarked by Alexansder Stasinski (thanks!), for $n=3$ (but not for higher $n$'s) the representation zeta function is explicitly computed in the Avni-Klopsch-Onn-Voll paper alluded to above (see Theorem E) and one should be able get out of it the exact numbers of irreducible representation of $\text{SL}_3(\mathbb{Z}/q\mathbb{Z})$ at each dimension, which is closer to answering the OP question. One can also hope to get a full answer using Kirillov orbit method. However, things get messy for higher $n$'s.

Answer 2

This computation could in principle be done using Clifford theory. Clifford theory tells you how to describe the representation theory of a group $G$ given that it can be described as an extension

$$1 \to N \to G \to H \to 1$$

and $SL_d(\mathbb{Z}/p^n\mathbb{Z})$ can be described as an extension

$$1 \to N \to SL_d(\mathbb{Z}/p^n \mathbb{Z}) \to SL_d(\mathbb{Z}/p^{n-1}\mathbb{Z}) \to 1$$

where $N$ is the kernel of the reduction $\bmod p^{n-1}$ map. It consists precisely of elements of $SL_d(\mathbb{Z}/p^n\mathbb{Z})$ congruent to $I \bmod p^{n-1}$, or equivalently of the form $I + p^{n-1} M$. Every such matrix is invertible over $\mathbb{Z}/p^n\mathbb{Z}$ and has determinant $1$ iff $\text{tr}(M) \equiv 0 \bmod p$, and multiplying two such matrices even shows that $N$ is isomorphic to the additive group of such matrices, hence

$$N \cong (\mathbb{Z}/p\mathbb{Z})^{d^2-1}.$$

If you like, you can think of $N$ as $\mathfrak{sl}_d(\mathbb{Z}/p\mathbb{Z})$.

Clifford theory simplifies substantially when $N$ is abelian, so this is helpful, although I think the detailed analysis will still be difficult to carry out, and doing it this way you'll have to repeat the analysis $n-1$ times to reduce to the case of $SL_d(\mathbb{Z}/p\mathbb{Z})$.

Answer 3

The answer to your stated question is certainly "no", which I can support indirectly by reference to the simpler case of rank 1 groups.

The basic question here is natural but is already quite difficult even in rank 1: study the representation theory of a given group scheme over a ring of $p$-adic integers (or more generally, the ring of integers of a local field) by working out the representations of the finite groups over finite residue class rings .

This type of question has a fairly long history, so it may be worthwhile to follow the paper trail. There are for example a number of relevant papers in the rank 1 case, including one by Kutzko here and a little later by A. Nobs and J. Wolfart here and here.

It's important here to keep an open mind about techniques not encountered in the study of matrix groups over finite fields. That study was done in a special case in the Simpson-Frame work (which has some minor errors), building on the recursive combinatorial determination of characters of finite general linear groups by Green in 1955. There was also the work of his student Srinivasan on $Sp_4$ and Chang-Ree on groups of type $G_2$ over finite fields, and ultimately the far more sophistiated work that flowed out of the 1976 Deligne-Lusztig construction of virtual characters. By now the theory over finite fields has acquired considerable detail through the efforts of Lusztig and others. As far as I can tell, there is still no overall program for constructing the representations (or even the characters) of the finite groups of Lie type over rings such as $\mathbb{Z}/q \mathbb{Z}$ which are not fields.