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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.

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2 Answers 2

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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 $2^5 = 32$.

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  • $\begingroup$ 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 $E_8$.) $\endgroup$
    – Noam D. Elkies
    Commented Sep 8, 2015 at 17:03
  • $\begingroup$ 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? $\endgroup$
    – YCor
    Commented Sep 9, 2015 at 8:37
  • $\begingroup$ Right, the $2^n$ dimensional representation of the automorphism group is projective but not linear. $\endgroup$
    – Noam D. Elkies
    Commented Sep 9, 2015 at 14:42
  • $\begingroup$ 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 $\endgroup$
    – Ian Gershon Teixeira
    Commented Mar 8, 2022 at 13:15
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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 $S$.

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