# On the finite simple groups with an irreducible complex representation of a given dimension

This answer of Geoff Robinson shows that a finite simple group admits an irreducible complex representation (irrep) of dimension $$3$$ if and only if it is isomorphic to $$A_5$$ or $$\mathrm{PSL}(2,7)$$.

Question: Is there such a classification for every dimension $$d$$?
[Or at least, for every small dimension, let's say less than $$15$$?]

Weaker question: Are there finitely many finite simple groups with an irrep of dimension $$d$$?

If yes: What are the best known lower/upper bounds for the order of a finite simple group with an irrep of dimension $$d$$?

The following table provides the number $$n$$ of finite simple groups $$G$$ of order less than $$10^6$$ with an irrep of a given dimension $$d < 15$$, together with the minimun and maximum orders.

$$\begin{array}{c|c|c|c|c} d &n&min&max & G \newline \hline 3 &2&60&168&A_5, \mathrm{PSL}(2,7) \newline \hline 4 &1&60&60&A_5 \newline \hline 5 &4 &60 &25920&A_5, A_6, \mathrm{PSL}(2,11), \mathrm{PSp}(4,3) \newline \hline 6 &4 &168 &25920&\mathrm{PSL}(2,7), A_7, \mathrm{PSU}(3,3), \mathrm{PSp}(4,3) \newline \hline 7 &5 &168 &20160&\mathrm{PSL}(2,7), \mathrm{PSL}(2,8), \mathrm{PSL}(2,13), \mathrm{PSU}(3,3), A_8 \newline \hline 8 &4 &168 &181440& \mathrm{PSL}(2,7), A_6, \mathrm{PSL}(2,8), A_9 \newline \hline 9 &4 &360 &3420& A_6, \mathrm{PSL}(2,8), \mathrm{PSL}(2,17), \mathrm{PSL}(2,19) \newline \hline 10 &5 &360 &25920& A_6, \mathrm{PSL}(2,11), A_7, M_{11}, \mathrm{PSp}(4,3) \newline \hline 11 &4 &660 &95040& \mathrm{PSL}(2,11), \mathrm{PSL}(2,23), M_{11}, M_{12} \newline \hline 12 &4 &660 &62400& \mathrm{PSL}(2,11),\mathrm{PSL}(2,13), \mathrm{PSL}(3,3), \mathrm{PSU}(3,4) \newline \hline 13 &5 &1092 &62400& \mathrm{PSL}(2,13), \mathrm{PSL}(3,3), \mathrm{PSL}(2,25), \mathrm{PSL}(2,27), \mathrm{PSU}(3,4) \newline \hline 14 &6 &1092 &604800& \mathrm{PSL}(2,13), A_7, \mathrm{PSU}(3,3), A_8, \mathrm{Sz}(8), J_2 \end{array}$$

• In such a setting it's quite natural to allow perfect central extensions of simple groups too (for instance $A_5$ occurs in $\mathrm{PSL}_2$, that is, some perfect central extension of $A_5$ occurs in $\mathrm{SL}_2$). (Also in certain exceptional cases, they occur in the form of some non-central abelian extensions, see the question Linear occurrences of finite simple groups)
– YCor
Commented Oct 20, 2019 at 19:35

I will write this an an answer, though the answer to the basic question is provided by one of the oldest results in finite group theory.

There is a theorem of C. Jordan, proved in the 19th century, that there is a function $$f : \mathbb{N} \to \mathbb{N}$$ such that whenever $$n \in \mathbb{N}$$ and $$G$$ is a finite subgroup of $${\rm GL}(n,\mathbb{C}),$$ then there is an Abelian subgroup $$A \lhd G$$ with $$[G:A] \leq f(n)$$.

Hence, in particular, for any integer $$d >1$$, there are only finitely many finite simple groups $$G$$ with a complex irreducible representation of degree $$d$$.

Explicit bounds for Jordan's theorem were given by many people over the years, such as Frobenius and Blichfeldt.

The bounds given prior to the classification of finite simple groups (CFSG) were far from optimal, but, using CFSG, B. Weisfeiler outlined a proof of a close to optimal bound in sufficiently large $$n$$.

Unfortunately, Weisfeiler disappeared before his work was published in complete form.

Recently, M.J. Collins published a complete proof with the optimal bound $$f(n) = (n+1)!$$ for sufficiently large $$n$$ ( I think $$n > 72$$ will do, if my memory is correct). I think Collins also gives the maximal value of $$f(n)$$ (with the optimal choice of function $$f$$) for smaller values of $$n$$, but this is not in general an answer to the first question.

I am not sure of the largest value of $$n$$ such that all finite irreducible simple subgroups of $${\rm GL}(n,\mathbb{C})$$ are known. I believe that people like G.Malle and G.Hiss have done this up to $$n =19$$ - ( Later edit- see Derek Holt's answer, which indicates a much higher bound due to those authors) ( the cases $$n \leq 11$$ were understood before CFSG by the work of many authors).

This is just to add some references to Geoff Robinson's answer for classifications of irreducible representations in low dimensions. These are all for quasisimple groups i.e. perfect groups $$G$$ for which $$G/Z(G)$$ is simple.

In the paper

G. Hiss and G. Malle. Low-dimensional representations of quasi-simple groups. LMS J. Comput. Math. 4 2001, 22-63. Corrigenda: LMS J. Comput. Math. 5 2002, 95-126,

the authors classify all such representations of degree up to $$250$$, not just in characteristic $$0$$, but in all characteristics other than the defining characteristic of finite groups of Lie type.

To complement this work, in the paper

F. Lübeck, Small degree representations of finite Chevalley groups in defining characteristic. LMS J. Comput. Math. 4 (2001), 135-169,

the author handles the case of groups of Lie type in defining characteristic, going up to dimension at least $$250$$ in all cases (and higher in many case).

These papers are all freely available online.

• See also Frank Lübeck's website, which in many cases goes beyond his 2001 paper: link. Commented Oct 20, 2019 at 15:48