Tensor Product of Krammer’s Representations of the P3

DOI: 10.4236/apm.2012.23026   PDF   HTML     3,727 Downloads   6,444 Views  

Abstract

We consider the complex specializations of Krammer’s representation of the pure braid group on three strings, namely K(q,t), where q and t are non-zero complex numbers. We then specialize the indeterminate t by one and replace by 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 .

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H. Tarraf and M. Abdulrahim, "Tensor Product of Krammer’s Representations of the P3," Advances in Pure Mathematics, Vol. 2 No. 3, 2012, pp. 190-194. doi: 10.4236/apm.2012.23026.

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1] M. N. Abdulrahim and M. Al-Tahan, “Complex Spe- cializa-tions of Krammer’s Representation of the Braid Group, B3,” Journal of Mathematics and Statistics, Vol. 4, No. 4, 2008, pp. 213-216. doi:10.3844/jmssp.2008.213.216
[2] M. N. Abdulrahim and M. Al-Tahan, “Krammer’s Rep- resentation of the Pure Braid Group, P3,” International Journal of Mathematics and Mathe-matical Sciences, Vol. 2010, 2010, Article ID 806502. doi:10.1155/2010/806502
[3] J. S. Birman, “Braids, Links and Mapping Class Groups,” Annals of Mathematical Studies, No. 8, Princeton University Press, Princeton, 1974.
[4] D. Krammer, “Braid Groups Are Linear,” Annals of Mathematics, Vol. 155, No. 1, 2002, pp. 131-156. doi:10.2307/3062152

  
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