Pensieve header: Showing the non-unitarity of Gassner on a pure w-braid.

Definitions (as in the paper).

(Subscript[U, i_])[t_] := ReplacePart[IdentityMatrix[n],
{{i,i}->1-t, {i,i+1}->1,
{i+1,i}->t, {i+1,i+1}->0}];
(Subscript[U, i_]^(-))[t_] := Inverse[(Subscript[U, i])[t]];
Ω[τ___] :=  Table[
Which[i<j, 0,i==j,(((1-(Subscript[t, {τ}[[i]]])))^(-1)),i>j, 1],
{i,n}, {j,n}];
(OverBar[X_]) := X /. (Subscript[t, i_]):>1/(Subscript[t, i]);
(Subscript[U, i_,j_]) := ReplacePart[IdentityMatrix[n],
{{i,i}->1, {i,j}->1-(Subscript[t, i]),
{j,i}->0, {j,j}->(Subscript[t, i])}];
(Subscript[V, i_,j_]):= ReplacePart[IdentityMatrix[n],
{{i,i}->1, {i,j}->1-(Subscript[t, j]),
{j,i}->0, {j,j}->(Subscript[t, i])}];
DD := DiagonalMatrix[Table[1-(Subscript[t, i]), {i,n}]];

Non unitarity.

n=3;

Hold[(OverBar[X_]) ]// FullForm

eigs=Eigenvalues[(Subscript[U, 1,2]).(Subscript[U, 3,2]).(Subscript[U, 2,1])]

eigs*OverBar[eigs] /. Table[(Subscript[t, i])->RandomReal[],{i,3}]