Advances in Pure Mathematics, 2012, 2, 190194 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, P3 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 nonzero complex numbers. We then specialize the indeterminate t by one and replace 3 ,:Kqt P 3, , GL ,1 q by q 33 123 :KqKq PGL 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 Bn 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 BAutF nn . Applying the free differential calculus to elements of n ut F sometimes gives rise to linear representations of Bn and its normal subgroup, the pure braid group de noted by Pn. The LawrenceKrammer representation named after Ruth Lawrence and Daan Krammer arises this way. Krammer’s representation is a repr esentation of the braid group Bn in 0 Aut V , wher 11 ,,t q GL me 12mnn and V0 is the free module of rank m over 11 , tq It is denoted by ,. qt For simplicity, we tead of ,. write K ins qt In previs work, we con sidered Krammer’s reprens of B3 and P3 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 AlTahan [2], a necessary and sufficient condition for the irreducibility of Krammer’s representation of degree three was found. However, in our cu rrent work, w e 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, Bn, 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 P3. In this way, we will have succeeded in constructing a representation of the pure braid group, P3, of degree nine and which is also irre ducible. Definition 1. [3] The br abstract group with presentation 111 ,, n iii 11 for 1,2, ,2,. if 1 iii n ij ji in B ij The generators 11 ,, 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 nerat ors o f B n. Definition 2. The ted by Pn, is the kernel of the group homomorphism . nn BS It consists of those braids which connect the f the left set to the ith item of the right set for all , 1<ijn C opyright © 2012 SciRes. APM
H. A. TARRAF, M. N. ABDULRAHIM 191 where 211 1 121121 . ijjjiiijj A Definition 3. [4] The image of ator each Artin generunder Krammer’s representation with respect to 1, ij ijn x the free basis of V0, is written as follows: 2,1 ,,1 ,,1; 1,,; ,; ,1; 1,1; r 1; 1. kk ik ik tq xikjk qxqxj ki k ik j kj 1 ,, 1 ,1 1, , ,1, , 2 ,,1 1, 1 1, 1, , o 1, ki ik kk kkk j kij kjk j ij ki ij kk xt q qxjk tqqxqxik k Kx xqx ik ijk xtqq xikk k ij j Now, we determine Krammer’s representation 11 ,:3,,Kqt PGLtq 3 using the Magnus representation of subgroups of the automorphisms group of free group with 12nn gen erators, where 11 ,tq is the ring o poly f Laurent variables. 24 23 2 00 11, 11 1 tq tq qqqq tq qqqq 23 24 2,3 24 11 1 11 00 qq qqtqq KAqqtq q tq and nomials on twoThe images of the generators under Krammer’s representation are as follows 1,2 KA 2 23 22 2 11 1 111 , 111 qtqq qqq t KAtqqtqtq qqqm tqqtq qqtqn 4 22 2 12211 11mqqqtqqqqqt 2 11111 .nqqtqqtq Specializing the indeterminates q and t to nonzero co if q For mo th 1,3 where nd a mplex numbers gives a representation 3 ,: 3,Kqt PGL which is irreduci 23 t ble if and only 3 2 1 ,1,1,1,1 and1.tqtq tqtq re details, see [2]. indeterminate t by one. We We now specialize the en write q for ,1 .Kq So, q is irreduci ble if and only 32 1and 1.qq By replacing t by one, the matrices of the generators be 4 3 2 00 11, 11 1 q qqq qq qqqq q 23 4 4 11 1 11 00 qqqqqq qq qq q 2 23 22 2 111 111 , 111 qqq qqq qAqqqqq qqm qqqq qqn K if come as follows: 1,2 qA K 2,3 qAK and 1,3 Copyright © 2012 SciRes. APM
H. A. TARRAF, M. N. ABDULRAHIM 192 where 1 1qqq and 2 11 .qq In our work, we consider the tensor product of two ir ducible complex specializations of Krammer’s repre tions of the pure braid group, namely, 12 . 4 22 1221 1mqqqqq 111nqqq re senta qKq 3. the iduciiof two ire ducible complexspecializations of Krammer’s represen tations of the pure braid group on three strings, P3. Theorem 1. For 2 1, 2 ()qq , the tensor product of two irreducible complex specializations of Krammer’s representations Sufficient Conditions for Irreducibility In this section, we find sufficient conditions that guaran teerreblity of the tensor product 33 123 :KqKq PGL is 2 121 4 3 ,1,qqqq irreducible if 44 4424 42 24 12121 2 424 3 1212 12 1, 1,, ,1,1,1,1. qqqqq q qqqqq q Proof. For simplicity, we write ij to mean 12ijij. qAq23 A by an invertib Let us diagonalize the matrix le matrix T, and conjugate the matrix 12 A by T. The invertible matrix T is given by 1212 1qa qqq 2 1 2 22 2 1112 1112 22 222 112 2 0 0 11 111 1111 000001 1 001000 00100100 000000001 aqc cqac d dq qqqqqqqq qc c qqq bbdde b qc d , 2 12 2 2 1 22 12 1 1 11 1 qqq q qq qa 11 11qa q a 0000 11bq T 0 0 where 22 12 1 22 11 22 1112 2 222 212 1 ,1, , 111 11 1 1and (1 )11 qq abqc qq qqqqq dq e qqq Diagonalizing 23 A by T, we get that 2 14 12 222 2 2 2 2 . q q 1 242 24 44 12 1000 00 0 00 0000 000 0 0000 000 0 0000 000 0 .0000 0 00 0000 00 00 0 0 0 0 q q q qq qq qq 23 0 A 12 4 2 12 0 000 00 0000 00 0 0000 00 0 q qq Copyright © 2012 SciRes. APM
H. A. TARRAF, M. N. ABDULRAHIM 193 Conjugating A12 by 12 =, ij aA where the entries of its first row and first column are given by T, we get that 22 2222 111122 11aq qq q 22 1 2 11,q q 2 22 322 12111122 11 1aq q qqqq 22 12 11,q q 22 2322 13 1122 11aqqq q 42 1 2 11,q q 222 12 11,q q 22 2 23 14112222 111aqqq q qq 22 2322 3 15112222 12 1111 11,aqqqqqqq qq 2 11 q 242 2322 3 16112222 12 111 1,aq qq q qqq q 1 22 24 2223 17 112212 1111,aq qqqqq 224 23 22 12 1 11,qq q 22 3 181111 11aq qqqq 22 44 2323 191122 12 1111aq qq qqq , 2 23 211 1122 21 1aqqqq 22 1 2 11,qqq 2 22 22 31 1 222 111,aqqqq 2 32 22 2 2212 21111,aqq qqqqq 41 1 1 22 22 2 2212 1111,qqqqqq 51 1 11 41 1aqq 22 2,q 2 61 12 22 2111 1aqqqq 222 22 71 1121 111,aq qqq 22 1 1 ,qq .q Supp Kq there ex the dimd eight inclusively. By convalent representation (after conjugation), we will easily see that this assumption leads to contradiction. Since the eigen values of 23 A are distinct by our hy 2 81 1 112 211 1aqqq 22 91 1 2 11aq ose, to get contradiction, that 33 123 :Kq PGL is reducible, then ists a proper nonzero invariant subspace S, where ension of S is between one an sidering such an equi pothesis, it follows that 12 ,,, s ll l See e where 1,, 9 , 18sl and l es are the standard un Case 1. 12 1,, . s ll l eeee Since 1dim8,S it vectors of 9. , it follows that there exists 12 ,,, s ill l ee e where 2, ,9i. Since 1 eSe , it follows that 12 1 eS which implies that 10 i a , a contradiction since 10 i a 2, ,9i by our hy pothesis. Case 2. 12 1,,,. s ll l eeee Since 1dim8,S it follows that there exists 12 ,,, s ill l eee e where 2,i ,9 and so Copyright © 2012 SciRes. APM
H. A. TARRAF, M. N. ABDULRAHIM 194 12 1 eS which implies that 10 i a, a contradiction since 10 i a 2, ,9i . REFERENCES [2] M. N. Abdulra. AlTahan, “Krammer’s Rep resentation of the Pure Braid Group, tional Journa 010, 20, Article ID 806502. doi:10.1155/2010/806502 [1] . A Repres f Mathematics and S No. 4, 2008, pp. 213216. doi:10.3844/jmssp.2008.2 M. N. Abdulrahim and MlTahan, “Complex Spe cializations of Krammer’s entation of the Braid Group, B3,” Journal otatistics, Vo 13.216 l. 4, him and MP,” Interna 3 l of Mathematics and Mathematical Sciences, Vol. 2 10 [3] J. S. Birman, “Braids, Links and Mapping Class Groups,” Annals of Mat, Princeton Univer sity Press, Prin hematical Studies, No. 8 ceton, 1974. [4] D. Krammer, “Braid Groups Are Linear,” Annals of M ath e matics, Vol. 155, No. 1, 2002, pp. 131156. doi:10.2307/3062152 Copyright © 2012 SciRes. APM
