### Abstract

The q-deformation of the chiral Gross-Taylor holomorphic string large N expansion of two dimensional SU(N) Yang-Mills theory is described. Delta functions on symmetric group algebras are replaced by the corresponding objects (canonical trace functions) for Hecke algebras. The role of the Schur-Weyl duality between unitary groups and symmetric groups is now played by q-deformed Schur-Weyl duality of quantum groups. For q -> 1, the string interpretation of Yang Mills theory relies on the description of Hurwitz spaces in terms of symmetric group data. While the algebraic deformation of the symmetric group data to Hecke algebra data is established in hep-th/0603056 , the geometrical story in terms of some q-deformation of Hurwitz spaces is an interesting open question.