Summary: 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 consider a linear representation of the braid group on three strings, namely, Krammer’s representation. The objective of our work is to study the irreducibility of a specialization of Krammer’s representation.
Approach: We specialize the indeterminates used in defining the representation to non zero complex numbers and work on finding invariant subspaces under certain conditions on the indeterminates.
Results: We find a necessary and sufficient condition that guarantees the irreducibility of Krammer’s representation of the braid group on three strings.
Conclusion: This is 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 , which might enable us to characterize all irreducible Krammer’s representations of various degrees.