Reducibility of the Cohen-Wales representation of the Artin group of type $D_n$

REDUCIBILITY OF THE COHEN–WALES REPRESENTATION OF THE ARTIN GROUP OF TYPE Dn

“…The generalized BMW algebra has the same generators as those of the Artin group in the same way the original BMW algebra has the same generators as those of the braid group. In [18], we use the tangle algebra defined in [5] to construct a representation of the BMW algebra of type D n BM W (D n ), which as a representation of the Artin group A(D n ) is equivalent to the generalized Lawrence-Krammer representation LK(D n ) introduced by Arjeh Cohen and David Wales in [4]. We use this representation to deduce a reducibility criterion for LK(D n ) as well as a conjecture which gives a criterion of semisimplicity for BM W (D n ).…”

“…Other constructions and proof of linearity can also be found in [8]. Reducibility criteria for these representations exist in types A and D (see [15] and [18] respectively). In each case, reducibility is shown for some complex specializations of the two parameters of the representation, while the representations are generically irreducible (see [5], [20], [26], [15] in type A and [5] and [18] in type D).…”

“…The problem in type A originated in Result (a) of [25], was further developed in Theorem 2 of [16] and was completely solved in Theorem B of [23], building upon [9]. The problem in type D is stated as a conjecture in [18] and is to this date not solved.…”

“…The rest of the proof that M is a right H-module for the action provided above is up to an appropriate change in the indices similar to the proof of Claim 1 of [18]. Recall that r 2 + m r = 1.…”

“…First, if we forget about node 1 and the edge joining nodes 1 and 3 on the Dynkin diagram, we obtain a Dynkin diagram of type D 5 on nodes 2, 3, 4, 5, 6. For the vectors indexed by the positive roots issued from this diagram, we use the same notations as in [18] with node i + 1 now playing the role of node i. So, for instance w 2j , j ≥ 4 (resp w 23 ) denotes the vector associated with the positive root α 2 + α 4 + • • • + α j (resp with the simple root α 2 ); we denote by w 3j , j ≥ 4 the vector associated with the positive root α 2 + α 3 + α 4 + • • • + α j ; we denote by w st , s ≥ 4 the vector associated with the positive roots…”