Faithfulness of the diagrammatic interpretation

Establish the faithfulness of the diagrammatic interpretation of the double affine Frobenius Hecke algebra H_n(A,z,q) by proving that the associated string diagrams on the thickened punctured torus represent distinct algebra elements whenever they are distinct in H_n(A,z,q).

Background

The paper gives a heuristic interpretation of monomial elements of H_n(A,z,q) as string diagrams on a thickened punctured torus, with arbitrary algebra elements represented by formal linear combinations of such diagrams. This framework is used to motivate the algebra’s symmetries and later constructions, including the Dehn-twist actions and the resulting outer SL_2(Z)-action.

Although the paper explains how the generators and defining relations can be represented diagrammatically, it does not establish that this representation is faithful. Proving faithfulness would show that the diagrammatic model captures the algebra without introducing additional identifications and would provide a rigorous foundation for extending diagrammatic methods to related Frobenius Hecke algebras and their applications.