Type $A$ DAHA and Doubly Periodic Tableaux

TL;DR

This paper constructs and classifies representations of the $GL_m$-type Double Affine Hecke Algebra at roots of unity using combinatorial objects called doubly periodic tableaux, linking algebraic and topological structures.

Contribution

It introduces a new class of graded representations parametrized by doubly periodic tableaux and proves they exhaust all graded $X$-semisimple representations, establishing faithfulness and topological interpretations.

Findings

Representations are parametrized by doubly periodic tableaux.

All graded $X$-semisimple representations are classified.

DAHA is realized as a skein algebra of the torus.

Abstract

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