Advances in Pure Mathematics, 2012, 2, 190-194 http://dx.doi.org/10.4236/apm.2012.23026 Published Online May 2012 (http://www.SciRP.org/journal/apm) Tensor Product of Krammer’s Representations of the Pure Braid Group, P 3 Hassan A. Tarraf, Mohammad N. Abdulrahim Department of Mathematics, Beirut Arab University, Beirut, Lebanon Email: {hat012, mna}@bau.edu.lb Received December 27, 2011; revised February 17, 2012; accepted February 25, 2012 ABSTRACT We consider the complex specializations of Krammer’s representation of the pure braid group on three strings, namely, where q and t are non-zero complex numbers. We then specialize the indeterminate t by one and replace   3 , : Kqt P   3, ,   GL   ,1 K q by   K q       3 3 1 2 3 : Kq Kq P GL      for simplicity. Then we present our main theorem that gives us sufficient conditions that guarantee the irreducibility of the tensor product of two irreducible complex specializations of Krammer’s repre- sentations . Keywords: Braid Group; Pure Braid Group; Magnus Representation; Krammer’s Representation 1. Introduction Let B n be the braid group on n strings. It has many kinds of linear representations. The earliest was the Artin rep- resentation, which is an embedding   B Aut F    n n . Applying the free differential calculus to elements of n Aut F sometimes gives rise to linear representations of B n and its normal subgroup, the pure braid group de- noted by P n . The Lawrence-Krammer representation named after Ruth Lawrence and Daan Krammer arises this way. Krammer’s representation is a representation of the braid group B n in     0 Aut V     , wher 1 1 , , t q    GL m e   1 2 m nn   and V 0 is the free module of rank m over 1 1 , . t q     It    is denoted by   , . K qt For simplicity, we tead of   , . write K ins K qt In previ s work, we con- sidered Krammer’s repre ns of B 3 and P 3 and we specialized the indeterminates to non zero complex numbers. We then found necessary and sufficient condi- tions that guarantee the irreducibility of such representa- tions. For more details, see [1,2]. Note that in a previous work of Abdulrahim and Al-Tahan [2], a necessary and sufficient condition for the irreducibility of Krammer’s representation of degree three was found. However, in our current work, we are dealing with a representation of higher degree (degree nine) and which also has two in- determinates. This made our work seem more difficult. For this reason, we had to be satisfied in this current work with only a sufficient condition for irreducibility, so we fell short of finding a necessary and sufficient condition for irreducibility. To make computations easier, we had to specialize the indeterminate t by one in order to have a one parameter complex specialization. In Section 2, we introduce the pure braid gr ou sentatio oup and K 2. Definitions aid group on n strings, B n , is the rammer’s representation. In Section 3, we present our main theorem, Theorem 1, which gives sufficient condi- tions that guarantee the irreducibility of the tensor prod- uct of two irreducible complex specializations of Kram- mer’s representations of P 3 . In this way, we will have succeeded in constructing a representation of the pure braid group, P 3 , of degree nine and which is also irre- ducible. Definition 1. [3] The br abstract group with presentation 1 1 1 , , n i i i       1 1 for 1, 2, , 2, . if 1 i i i n i j j i i n B i j                         The generators 1 1 , , n      are called the standard ge Pure braid group on n strands, de- no ith item o i. It is generated by the generators ij nerators of B n . Definition 2. The ted by P n , is the kernel of the group homomorphism . n n B S  It consists of those braids which connect the f the left set to the ith item of the right set for all , 1 < i j n   A Copyright © 2012 SciRes. APM