Krammer's Representation of the Pure Braid Group,

We consider Krammer's representation of the pure braid group on three strings: , where and are indeterminates. As it was done in the case of the braid group, , we specialize the indeterminates and to nonzero complex numbers. Then we present our main theorem that gives us a necessary and sufficient condition that guarantees the irreducibility of the complex specialization of Krammer's representation of the pure braid group, . 1. Introduction Let be the braid group on strings. There are a lot of linear representations of . The earliest was the Artin representation, which is an embedding , the automorphism group of a free group on generators. Applying the free differential calculus to elements of sometimes gives rise to linear representations of and its normal subgroup, the pure braid group denoted by [1]. The Burau, Gassner, and Krammer's representations arise this way. In a previous paper, we considered Krammer's representation of the braid group on three strings and we specialized the indeterminates to nonzero complex numbers. We then found a necessary and sufficient condition that guarantees the irreducibility of such a representation. For more details, see [2]. In Section 2, we introduce some definitions of the pure braid group and Krammer's representation. In Sections 3 and 4, we present our work that leads to our main theorem, Theorem 4.2, which gives a necessary and sufficient condition for the specialization of Krammer's representation of to be irreducible. 2. Definitions Definition 2.1 (see [1]). The braid group on strings, , is the abstract group with presentation for if . The generators are called the standard generators of . Definition 2.2. The kernel of the group homomorphism is called the pure braid group on strands and is denoted by . It consists of those braids which connect the th item of the left set to the th item of the right set, for all . The generators of are , where . Let us recall the Lawrence-Krammer representation of braid groups. This is a representation of in , where and is the free module of rank over . The representation is denoted by . For simplicity we write instead of . What distinguishes this representation from others is that Krammer's representation defined on the braid group, , is a faithful representation for all [3]. The question of whether or not a specific linear representation of an abstract group is irreducible has always been a significant question to answer, especially those representations of the braid group and its normal subgroups. In a previous result, we determined a necessary and sufficient condition for