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It is known that the Lawrence-Krammer representation of the Artin group oftype $A_{n-1}$ based on the two parameters $t$ and $q$ that was used by Krammerand independently by Bigelow to show the linearity of the braid group on $n$strands is generically irreducible. Here, we recover this result and showfurther that for some complex specializations of the parameters therepresentation is reducible. We give all the values of the parameters for whichthe representation is reducible as well as the dimensions of the invariantsubspaces. We deduce some results of semisimplicity of theBirman-Murakami-Wenzl algebra of type $A_{n-1}$.

Irreducibility of the Lawrence-Krammer representation of the BMW algebra of type $A_{n-1}$