  Lawrence-Krammer representations are an important family of linear
representations of Artin-Tits groups of small type, which are known, under some
assumptions on the parameters, to be faithful when the type is spherical (or
more generally when they are restricted to the Artin-Tits monoid) and
irreducible when the type is connected.
  Here, we investigate an analogue of these representations --- introduced by
Digne in the spherical cases --- for every Artin-Tits monoid that appears as
the submonoid of fixed points of an Artin-Tits monoid of small type under a
group of graph automorphisms, and for the corresponding Artin-Tits group.
  Under the same assumptions on the parameters as in the small type cases, we
first show that these so-called "twisted Lawrence-Krammer representations" are
faithful, and we then prove, by computing their formulas when the group of
graph automorphisms is of order two or three, their irreducibility in all the
spherical and connected cases but one.
