Tensor products of specializations of the Burau representation

Abstract

The reduced Burau representation is a one-parameter representation of B n , the braid group on n strings. Specializing the parameter to a nonzero complex number y gives a representation n (y) : B n → GL(C n−1 ) which is either irreducible or has an irreducible composition factorˆ n (y) : B n → GL(C n−2 ). We prove that the tensor product of an irreducible n (y) orˆ n (y) with an irreducible n (z) orˆ n (z) is irreducible unless y = z ±1 .

Tensor Products of the Gassner Representation of The Pure Braid Group†

“…Our work is an extension of a previous one, where we have proved in [3] that the representation obtained, by tensoring irreducible complex specializations of the Burau representation, namely, n (x) or n (x) and n (y) or n ( y) is irreducible if and only if x y.…”

“…The proof is the same as in [3]. By Proposition 1, A n 1 A n 1 is the unique minimal nonzero U j -submodule of n 1 n 1 .…”

“…In [3], we considered the tensor product of complex specializations of the Burau representation of the braid group and found a necessary and sufficient condition that guarantees irreducibility. Here, we consider the tensor product of the following irreducible representation restricted to a free normal subgroup of the pure braid group, namely, U j :…”

“…The idea of the proof is similar to that in [3]. We consider a free normal subgroup of the pure braid group of rank n 1 denoted by Uj where 1 j n. Let [Uj] be the group algebra of Uj over , and let Abe the augmentation ideal of…”

On the irreducibility of linear representations of the pure braid group

“…Proof. The proof is similar to that in [2]. Here, we will take the free normal subgroup, U r , of rank n − 1.…”