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Complex specializations of Krammer’s representation of the braid group, $B_3$. (English) Zbl 1188.20030
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 $n$, which might enable us to characterize all irreducible Krammer’s representations of various degrees.

20F36Braid groups; Artin groups
20C15Ordinary representations and characters of groups
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