Faithful Burau-like representations of some rank two Garside groups and torus knot groups

Abstract We give a method to produce faithful representations of the groups G ( n , m ) = ⟨ X , Y | X m = Y n ⟩ $G(n,m)=\langle X, Y \ \vert \ X^m = Y^n \rangle $ upper G left parenthesis n comma m right parenthesis equals left angle bracket upper X comma upper Y vertical bar upper X Superscript m Baseline equals upper Y Superscript n Baseline right angle bracket in GL 2 ( C [ t ± 1 , q ± 1 ] ) $\mathrm {GL}_2(\mathbb {C}[t^{\pm 1}, q^{\pm 1}])$ upper G upper L 2 left parenthesis double struck upper C left bracket t Superscript plus or minus 1 Baseline comma q Superscript plus or minus 1 Baseline right bracket right parenthesis . These groups are Garside groups and the Garside normal forms of elements of the corresponding monoid can be explicitly recovered from the matrices, in the spirit of Krammer’s proof of the linearity of Artin’s braid groups. We use this method to construct several explicit faithful representations of the above groups, among which a representation which generalizes the reduced Burau representation of B 3 ≅ G ( 2 , 3 ) $B_3 \cong G(2,3)$ upper B 3 approximately equals upper G left parenthesis 2 comma 3 right parenthesis to a large family of groups of the form G ( n , m ) $G(n,m)$ upper G left parenthesis n comma m right parenthesis with n , m coprime (which are torus knot groups). Like the Burau representation, this representation specializes to a representation of a reflection-like quotient that we previously introduced, called 2-toric reflection group . As a byproduct, we get a “Burau representation” for some exceptional complex braid groups, which also shows that the latter embed into their Hecke algebra.