These are the Mathematica solutions for the paper:

“Permutation-type solutions to the Yang-Baxter and other n-simplex equations” by J. Hietarinta.

computations by Dara O Shayda

dara@lossofgenerality.com

<<Notation`

This formulation is incomprehensible and I guessed how it works:

Notation[(NotationTemplateTag[(Subscript[R, i_,j_])]) ⟹ (NotationTemplateTag[makeA[A,{i_,j_},3]])]
Notation[(NotationTemplateTag[(Subscript[R, i_,j_,k_])]) ⟹ (NotationTemplateTag[makeA[A,{i_,j_,k_},6]])]
Notation[(NotationTemplateTag[(Subscript[R, i_,j_,k_,h_])]) ⟹ (NotationTemplateTag[makeA[A,{i_,j_,k_,h_},10]])]

makeMatrix[dim_, a_]:=Array[Subscript[a,#1,#2]&,dim]

(* makeA generates the A matrices in Hietarinta paper *)

makeA[A_, list_,n_]:=Module[{res, i,j,k},

res=ConstantArray[-1,{n,n}];

Table[If[MemberQ[list, i]!= True, res [[i]][[j]]=KroneckerDelta[i,j]; res [[j]][[i]]=KroneckerDelta[j,i] ], {i, 1, n}, {j, 1, n}];
 
Table[k=1;Table[ If[res[[list[[i]]]][[j]] == -1, res[[list[[i]]]][[j]] = A[[i]][[k]];k=k+1];,{j,1,n}],{i , 1, Length[list]}];
res

]

4-simplex

A=makeMatrix[{4,4},a];
A//MatrixForm

Make the R matrices:

(Subscript[R, 1,2,3,4])  //MatrixForm

(Subscript[R, 1,5,6,7])  //MatrixForm

(Subscript[R, 2,5,8,9])  //MatrixForm

(Subscript[R, 3,6,8,10])  //MatrixForm

(Subscript[R, 4,7,9,10])  //MatrixForm

Too large set of equations of 3rd degree to printout:

((Subscript[R, 1,2,3,4]).(Subscript[R, 1,5,6,7]).(Subscript[R, 2,5,8,9]).(Subscript[R, 3,6,8,10]).(Subscript[R, 4,7,9,10])-(Subscript[R, 4,7,9,10]).(Subscript[R, 3,6,8,10]).(Subscript[R, 2,5,8,9]).(Subscript[R, 1,5,6,7]).(Subscript[R, 1,2,3,4]))//MatrixForm

sol=Solve[(Subscript[R, 1,2,3,4]).(Subscript[R, 1,5,6,7]).(Subscript[R, 2,5,8,9]).(Subscript[R, 3,6,8,10]).(Subscript[R, 4,7,9,10])-(Subscript[R, 4,7,9,10]).(Subscript[R, 3,6,8,10]).(Subscript[R, 2,5,8,9]).(Subscript[R, 1,5,6,7]).(Subscript[R, 1,2,3,4]) == ConstantArray[0,{10,10}], Flatten[A]];

sol2= DeleteDuplicates[sol];
Length[sol2]

Note the variables for the solutions:

Flatten[A]

I see none of thes solutions or even hint from Hietaina paper:

Table[ A/.sol2[[i]] // MatrixForm , {i, 1, Length[sol2]}]