# Linear occurrences of finite simple groups

Let $S$ be a finite simple group. All representations below are over the complex numbers. Let

• $d_0(S)$ be the smallest dimension of a faithful representation of $S$,
• $d_1(S)$ be the smallest dimension of a faithful representation of some central extension of $S$, and
• $d_2(S)$ be the smallest dimension of a faithful representation of some finite group admitting $S$ as quotient (or, equivalently, of some finite group admitting $S$ as Jordan-Hölder factor).

In a sense, $d_2(S)$ can be thought of the smallest dimension in which $S$ occurs linearly. On the other hand, $d_0$ and $d_1$ are well-documented, but, as far as I know, not $d_2$.

Trivially $d_0\ge d_1\ge d_2$, and the left-hand inequality can be strict: $d_0(\mathrm{Alt}_5)=3>2=d_1(\mathrm{Alt}_5)$.

Does there exist $S$ with $d_1(S)>d_2(S)$?

I'd be interested as well by any information about $d_2$, including values for small non-abelian simple groups, or some particular families.

Note: in a naive attempt to show $d_1=d_2$, we could wonder if any finite group having $S$ as quotient admits a subgroup isomorphic to some central extension of $S$; a counterexample is pointed out by Derek Holt here.

Seems that $S = O_{2n}^\pm(2)$ are examples of this for $n=5$, and probably for all $n \geq 5$. Such $S$ is a Jordan-Hölder factor of the automorphism group of the extraspecial group $2^{1+2n}_\pm$, so $d_2 \leq 2^n$. But the Schur multiplier is trivial, so $d_0 = d_1$, and the ATLAS of Conway et al. reports minimal faithful representations of dimensions $154$ for $O_{10}^-(2)$ and $155$ for $O_{10}^+(2)$, both larger than $2^5 = 32$.

• Also $d_0=d_1=34$ for $O_8^-(2)$ but $d_2 \leq 2^4 = 16$. (But for $O_8^+(2)$ we run into the Weyl group of $E_8$.) Sep 8, 2015 at 17:03
• If I understand correctly, some central extension of order 2 of $\mathrm{Aut}(2^{1+2n}_\pm)$ (rather than $\mathrm{Aut}(2^{1+2n}_\pm)$ itself) admits a faithful representation of dimension $2^n$, is that correct?
– YCor
Sep 9, 2015 at 8:37
• Right, the $2^n$ dimensional representation of the automorphism group is projective but not linear. Sep 9, 2015 at 14:42
• The faithful representation of $2_+^{1+2n}.O_{2n}^+(2)$ (the order 2 central extension of $Aut(2_+^{1+2n})$ which you mentioned) under consideration here can be realized very concretely as the normalizer in the real orthogonal group $O_{2^n}(\mathbb{R})$ of the subgroup generated by tensor powers of $X=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ and $Z=\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}$. In other words $2_+^{1+2}=<X,Z>$ and $2_+^{1+4}=<X\otimes X ,X \otimes Z, Z \otimes X, Z \otimes Z>$. See for example Self Dual Codes and Invariant Theory ch.6 real Clifford group Mar 8, 2022 at 13:15

Searching and starting from Noam's answer, I found some extra information:

the phenomenon pointed out by Noam on automorphism groups of extraspecial 2-groups seems to appear in: R.L. Griess, Jr. Automorphisms of extra special groups and nonvanishing degree 2 cohomology. Pacific J. Math 48 (73) 403-422. (Link, MR link)

The discussion about $d_1$ and $d_2$ seems to appear in: W. Feit and J. Tits, Projective representations of minimum degree of group extensions, Canad. J. Math. 30 (1978) 1092-1102, and continues in P. Kleidman and M. Liebeck. On a theorem of Feit and Tits, Proc. AMS 107(2), 1989, 315-322. (Link, MR link)

Feit-Tits establish that if $d_2(S)<d_1(S)$, then $S$ is of Lie type over a finite field of characteristic 2, and in this case $d_2$ is a power of 2. Kleidman and Liebeck establish the precise list of those such groups of Lie type that indeed satisfy $d_2<d_1$, including the computation of $d_2$.

In particular, it follows from this classification that the smallest dimension in which some finite simple groups occur" but not through a projective representation is 16, namely for the groups $\Omega_8^-(2)$, $\mathrm{Sp}_8(2)$, $\mathrm{Sp}_4(4)$.

Besides, if we consider representations over an algebraically closed field of characteristic $p\ge 5$, defining $d_i^{(p)}$ in the same fashion: the picture is exactly the same. In char. 3 it's essentially the same up to a minor difference in the classification among those $S$ of Lie type over a field of characteristic 2. On the other hand in characteristic 2 it's dramatically different: $d_1^{(2)}(S)=d_2^{(2)}(S)$ for all $S$.