Question

It is known that for a simple algebraic group over an algebraically closed field of positive characteristic (which I assume to be {\it good} for the group), the Weyl modules corresponding to the fundamental weights are irreducible if the group is either $SL_n$ or $SO_n$ (Wong 71). For the symplectic group, this is not true for an arbitrary fundamental weight. Foulle provides certain conditions on the weights satisfying which the corresponding Weyl modules of the symplectic group are irreducible. Does one know anything about the irreducibility of the fundamental Weyl modules for the exceptional groups? I believe its true for $G_2$ by results of Premet, Humphreys and others. Is it true also for other exceptional groups?

Answer 1

It is true in types $G_2$, $F_4$ and $E_6$, but in types $E_7$ and $E_8$ one has to exclude $p=7$, $p=13$ and $p=19$ as well (I think $p=19$ occurs only in type $E_8$). More on this can be found in Jantzen's paper "First cohomology groups for classical Lie algebras" published in Progress in Math., vol. 95, 1991 (the best way to find this paper is to google the title). For related results, see also the paper by Gilkey and Seitz titled "Some representations of exceptional Lie algebras" and published in Geom. Dedicata, 25, (1988), 407-416.

That the Weyl module $V(\varpi_6)$ for $G=E_7(K)$ is reducible in characteristic $7$ is quite easy to see directly: the minuscule module $V(\varpi_7)$ has a nondegenerate $G$-invariant symplectic form and by using Weyl's formula one observes that $V(\varpi_6)$ can be obtained from the second fundamental Weyl module for $Sp_{56}(K)$ by restriction to $G$. The latter Weyl module is reducible in characteristic $7$ as $7$ divides $56$.

Answer 2

[EDIT] My original incomplete answer is below, but Sasha Premet has provided the complete reference, the table of 4.6 in Jantzen's 1991 paper. Jantzen recalls that he computed various cases in types $E_\ell$ not covered by the computer calculations of his Oregon colleagues Gilkey and Seitz, using his amazing Sum Formula. This formula applies for all primes but is usually quite daunting to compute. Probably his non-computer results are completely correct, but skeptics should feel free to duplicate them by whatever means. It's definitely challenging to get any conceptual insight for these small primes. (Revisions to my lecture notes are posted here.)

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This question probably hasn't been fully answered for the exceptional types $E_6, E_7, E_8$. In general, the linkage principle (developed fully in Jantzen's book Representations of Algebraic Groups) ensures that for $p$ "large enough", each fundamental weight $\varpi$ lies in the closure of the lowest $p$-alcove for the affine Weyl group (of Langlands dual type) and thus the Weyl module $V(\varpi)$ is indeed simple. Here "large enough" depends on the Coxeter number and the coefficients of the highest short root, so the problem reduces to a computational one for a definite range of primes. The bad primes (possibly 2, 3, 5 for exceptional types) cause special problems, but apart from those the fundamental modules for types $G_2, F_4$ are well-behaved.

In Chapter 4 of my 2006 survey Modular Representations of Finite Groups of Lie Type (LMS Lecture Notes Series 326), I tried to cover all results known to me, with complete references. For example, Table 1 on page 37 summarizes results computed by Gilkey-Seitz for $F_4$ which confirm that fundamental Weyl modules are simple when $p>3$. Symplectic groups have been thoroughly examined by Premet-Suprunenko, Foulle, McNinch. And so on.

The fundamental modules for types $E_\ell$ probably haven't all been studied well enough to settle all cases. However, the few fundamental weights which are minuscule pose no problem since the weights of the corresponding Weyl modules form a single orbit under the Weyl group and thus the modules remain simple for all $p$. But otherwise the linkage principle just gives a bound on how much computation is needed. Probably nothing unusual happens for good primes, but that's not clearly documented yet.

One other theoretical remark: Lusztig's conjecture should handle all primes greater than the Coxeter number in a uniform way, but it has not yet been proved for this optimal lower bound. Meanwhile direct computation using existing algorithms (which go back to early work by Wong and Burgoyne) seems to be the only alternative.

Answer 3

You might also find the data on Frank Luebeck's website useful:

http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/WMSmall/index.html