Parameters for which the Lawrence–Krammer representation is reducible

Abstract

We show that the representation, introduced by Lawrence and Krammer to show the linearity of the braid group, is generically irreducible. However, for some values of its two parameters when these are specialized to complex numbers, it becomes reducible. We construct a representation of degree n(n−1) 2 of the BMW algebra of type A n−1 . As a representation of the braid group on n strands, it is equivalent to the Lawrence-Krammer representation where the two parameters of the BMW algebra are related to those appearing in the Lawrence-Krammer representation. We give the values of the parameters for which the representation is reducible and give the proper invariant subspaces in some cases. We use this representation to show that for these special values of the parameters, the BMW algebra of type A n−1 is not semisimple.

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.…”

“…This number is the same as the dimension of the Specht module S µ , where S µ denotes a class of irreducible H F,r 2 (n)-module for each partition µ of n. By Corollary 2 of [22], when H F,r 2 (n) is semisimple, the irreps of H F,r 2 (n) have degree 1, n − 1,…”

Invariant subspaces of the Lawrence–Krammer representation

“…Proof. This is Theorem 5 of [13]. In particular, it is shown that for n ≥ 3 and n = 4, S (n−1,1) occurs in the L-K space V (n) for l = 1 r n−3 and for l = − 1 r n−3 , while the conjugate Specht module S (2,1 n−2 ) cannot occur in the L-K space.…”

“…When the representation ν (n) is reducible, the action on a proper invariant subspace of V (n) is an Iwahori-Hecke algebra action: this is Proposition 1 of [13]. The following two theorems stated here for the Iwahori-Hecke algebra of the symmetric group Sym(n) instead of the symmetric group Sym(n) are due to James in [7].…”

“…We now recall some results of [13] about the existence of a one-dimensional invariant subspace of the L-K space and of an irreducible (n − 1)-dimensional invariant subspace of the L-K space for some values of the parameters l and r. when l = 1 r 3 , it is spanned by w 12 + r w 13 + r 2 w 23 when l = −r 3 , it is spanned by w 12 − 1 r w 13 + 1 r 2 w 23 If r 6 = −1, there are exactly two one-dimensional invariant subspaces of V (3) and they are respectively spanned by the vectors above.…”