Irreducibility of the Lawrence–Krammer representation of the BMW algebra of type An−1

Abstract

The Lawrence-Krammer representation introduced by Lawrence and Krammer in order to show the linearity of the braid group is generically irreducible. We show this fact and show further that for some values of its two parameters, when these are specialized to complex numbers, the representation becomes reducible. We describe what these values are and give a complete description of the dimensions of the invariant subspaces when the representation is reducible. To cite this article: C. Levaillant, C. R. Acad. Sci. Paris, Ser. I 347 (2009). RésuméIrréductibilité de la représentation de Lawrence-Krammer de l'algèbre BMW de type A n−1 . La représentation de Lawrence-Krammer, introduite par Lawrence et Krammer pour montrer la linéarité du groupe de tresses, est génériquement irréductible. On montre ce fait et on montre également que lorsque les deux paramètres de la représentation prennent certaines valeurs complexes, la représentation devient réductible. On donne ici toutes les valeurs des paramètres pour lesquelles la repré-sentation est réductible, ainsi que les dimensions des sous-espaces stables. Pour citer cet article : C. Levaillant, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

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

“…Some of these powers are identified twenty years later in the Ph.D. thesis of [20] and Theorem 2, point (ii) of the present paper can be viewed as a generalization of Theorem 2 of [22]. We recall below this result in type A.…”

“…In [20], we found matrix representations (P i ) 1≤i≤4 and (Q i ) 1≤i≤4 of degree 5 for respectively S (3,2) and S (2,2,1) . This is Fact 1 page 77 of [20]. These are matrix representations of H F,r 2 (5).…”

“…Equality (18) can be obtained by first using the relation e 2 n = δ e n , then applying the relation e i e j e i = e i for adjacent nodes i and j multiple times, finally by using the fact that e 1 and g 2 commute and applying e 2 1 = δ e 1 . To get (20), observe that e 1 e 2 = 0 in M n , then apply the same machinery as before. Now 1 δ e n acts to the right like the identity on any word ending in e n .…”

“…We use our representation to give a reducibility criterion for the Cohen-Wales representation of type D n . Our work can be viewed as a generalization of [21], where a reducibility criterion is given for the Lawrence-Krammer representation of the braid group.…”

“…By § 3.2, except when n = 4 and when n = 6, there are exactly two inequivalent irreducible representations of H(D n ) of degrees (n − 1). In [20], page 53, we provide matrix repre-…”