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I am interested in reducing reflection representations of complex reflection groups modulo a prime $p$. For the infinite family $G(m,r,n)$, it is straightforward to get "good reduction" provided that $p$ does not divide $m$.

For the exceptional groups, however, there are many descriptions of explicit matrix representations. One such is in Shephard-Todd's original paper, and some others are provided by the software packages MAGMA and CHEVIE. However, sometimes these descriptions aren't optimal. For example, they might involve $\sqrt{3}$ when it really isn't needed (so that reducing mod 3 will result in bad reduction). I discovered this for some small examples by experimentation, but it's hard to do this systematically. I am aware of the paper "Schur indices and splitting fields of the unitary reflection groups" by Mark Benard which states which algebraic integers are needed for a particular group.

So my question: given an exceptional reflection group $G$ and a prime $p$, is there a good reference to finding a reflection representation of $G$ in characteristic $p$ that has the same dimension as that of characteristic 0? I am willing to allow the center to be killed when the characteristic divides its order, but otherwise it would be nice to have it be faithful. It would be nicer to have a single form that can be reduced mod a prime, but I am also willing to just have a bunch of ad hoc examples for small primes.

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Have you satisfied yourself already about the behavior of real reflection groups and their reflection representations? This is usually the first place one would notice uniformity or lack of it relative to reduction mod primes, though I don't have all the literature at hand to check. – Jim Humphreys Jun 12 '11 at 22:39

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