# What is known about 2-modular representations of Ree groups of type $F_4$?

A too-vaguely worded question posted today about Suzuki and Ree groups reminds me to revisit a concern I never followed up years ago when assembling information about modular representations of finite groups of Lie type in the defining characteristic. One extreme case, for which basic data still seems incomplete, is the Ree group of type $\mathrm{F}_4$ in characteristic $p=2$. Here the extra-twisted finite simple groups live only over finite fields of order an odd power of $p$ starting with $2^3$. Often the group is denoted $^2\!\mathrm{F}_4(q^2)$ with $q^2 = 2^{2r+1}$ with $r \geq 1$, to make formulas for order and the like resemble those for the usual type $\mathrm{F}_4$ Chevalley groups.

While the older computer work by Gilkey-Seitz gives good results on simple modules for $\mathrm{F}_4$ for some small primes and small highest weights, it seems to leave some guesswork about the 16 restricted representations for $p=2$. These in turn feed into the somewhat more complicated recipe found by Steinberg for the Ree groups. Beyond this there are the (indecomposable) projective covers of those simples, along with their relationship to ordinary characters (which are known in principle from the Deligne-Lusztig viewpoint).

Are there more complete results about simple (and projective) modules for the Ree groups, at least for small powers of 2?

While these are extreme in terms of other groups of Lie type, they form a natural part of the modular theory and should be computable at least in the smallest cases.

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The web page web.mat.bham.ac.uk/atlas/v2.0/exc/TF42 is dated 1999 and gives information on the first group. It says all irreducible representations over $\mathbb{F}_2$ are known. –  Bruce Westbury Jan 25 '13 at 8:15
@Bruce: This group is usually called the Tits group (as the amusinmg apology in your link doesn't quite say). The excluded Ree group with $r=0$ isn't simple, but Tits observed that its derived group is a new simple group. Veldkamp worked out the 2-modular simples, whose Brauer characters are included in An Atlas of Brauer Characters. (References in 20.5 of my LMS lecture notes.) The Ree groups are harder to get at in terms of detail, which I'm trying to pin down. –  Jim Humphreys Jan 26 '13 at 0:02