Complex Specializations of Krammer's Representation of the Braid Group, B₃

Abstract

Problem statement: Classifying irreducible complex representations of an abstract group has been always a problem of interest in the field of group representations. In our study, we considered a linear representation of the braid group on three strings, namely, Krammer's representation. The objective of our work was to study the irreducibility of a specialization of Krammer's representation. Approach: We specialized the indeterminates used in defining the representation to non zero complex numbers and worked on finding invariant subspaces under certain conditions on the indeterminates. Results: we found a necessary and sufficient condition that guarantees the irreducibility of Krammer's representation of the braid group on three strings. Conclusion: This was a logical extension to previous results concerning the irreducibility of complex specializations of the Burau representation. The next step is to generalize our result for any n, which might enable us to characterize all irreducible Krammer's representations of various degrees.