Calculating with Wajnryb’s element

Inputting the basic complex reflections

We determine the formulas for these elements by way of Corollary 4.4, using the normalization conventions established in Section 5.1.

M0 = Transpose[{{λ,0,0,0,0,0},{0,1,0,0,0,0},{0,0,1,0,0,0},{-λ,0,0,1,0,0},{0,0,0,0,1,0},{0,0,0,0,0,1}}];
M1 = Transpose[{{1,0,0,0,0,0},{0,λ,0,0,0,0},{0,-λ,1,0,0,0},{0,0,0,1,0,0},{0,0,0,0,1,0},{0,0,0,0,0,1}}];
M2 = Transpose[{{1,0,0,0,0,0},{0,1,1,0,0,0},{0,0,λ,0,0,0},{0,0,-λ,1,0,0},{0,0,0,0,1,0},{0,0,0,0,0,1}}];
M3 = Transpose[{{1,0,0,1,0,0},{0,1,0,0,0,0},{0,0,1,1,0,0},{0,0,0,λ,0,0},{0,0,0,-λ,1,0},{0,0,0,0,0,1}}];
M4 = Transpose[{{1,0,0,0,0,0},{0,1,0,0,0,0},{0,0,1,0,0,0},{0,0,0,1,1,0},{0,0,0,0,λ,0},{0,0,0,0,-λ,1}}];
M5 = Transpose[{{1,0,0,0,0,0},{0,1,0,0,0,0},{0,0,1,0,0,0},{0,0,0,1,0,0},{0,0,0,0,1,1},{0,0,0,0,0,λ}}];

Defining W

Next we define W, following Section 5.1.

B = M3.M2.M4.M3.M1.M5.M2.M4.M3;
M0prime = B.M0.Inverse[B];
W = (M0.M0prime.M0).Inverse[M0prime.M0.M0prime];

Finding the characteristic polynomial

CharacteristicPolynomial[W,x]//Simplify

Ruling out roots of unity

First, we isolate the expression P(λ) appearing as the linear coefficient in the quadratic factor of the characteristic polynomial.

P[λ_]:=(1/(λ^9))(-1+3 λ-(λ^2)-8 (λ^3)+13 (λ^4)+(λ^5)-23 (λ^6)+20 (λ^7)+12 (λ^8)-34 (λ^9)+12 (λ^10)+20 (λ^11)-23 (λ^12)+(λ^13)+13 (λ^14)-8 (λ^15)-(λ^16)+3 (λ^17)-(λ^18));

Next, we plot  on the range  and verify that   on this range.

Plot[Re[P[Exp[I*θ]]]+2,{θ,Pi/2,4*Pi/5}]