Type A DAHA and doubly periodic tableaux

Analogously to the construction of Suzuki and Vazirani, we construct representations GLm-type Double Affine Hecke Algebra at roots unity. These are graded weight spaces for X-variables parametrized by combinatorial objects call doubly periodic tableaux. We show that our exhaust all X-semisimple representations, direct sum is faithful. Jordan Vazirani rectangular DAHA can be interpreted in terms ribbon fusion categories associated Uq(glN) Combining structure with faithfulness deduce a conjecture Morton Samuelson about realization as skein algebra torus base string modulo certain local relations.