×

Higher dimensional irreducible representations of the pure braid group. (English) Zbl 1267.20055

Summary: The reduced Gassner representation is a multi-parameter representation of \(P_n\), the pure braid group on \(n\) strings. Specializing the parameters \(t_1,t_2,\dots,t_n\) to nonzero complex numbers \(x_1,x_2,\dots,x_n\) gives a representation \(G_n(x_1,\dots,x_n)\colon P_n\to\mathrm{GL}(\mathbb C^{n-1})\) which is irreducible if and only if \(x_1\cdots x_n\neq 1\). In a previous work, we found a sufficient condition for the irreducibility of the tensor product of two irreducible Gassner representations. In our current work, we find a sufficient condition that guarantees the irreducibility of the tensor product of three Gassner representations. Next, a generalization of our result is given by considering the irreducibility of the tensor product of \(k\) representations (\(k\geq 3\)).

MSC:

20F36 Braid groups; Artin groups
20C15 Ordinary representations and characters
PDFBibTeX XMLCite
Full Text: DOI Link