  We introduce the basics of the nonabelian duality transformation of SU(N) or
U(N) vector-field models defined on a lattice. The dual degrees of freedom are
certain species of the integer-valued fields complemented by the symmetric
groups' \otimes_{n} S(n) variables. While the former parametrize relevant
irreducible representations, the latter play the role of the Lagrange
multipliers facilitating the fusion rules involved. As an application, I
construct a novel solvable family of SU(N) D-matrix systems graded by the rank
1\leq{k}\leq{(D-1)} of the manifest [U(N)]^{\oplus k} conjugation-symmetry.
Their large N solvability is due to a hidden invariance (explicit in the dual
formulation) which allows for a mapping onto the recently proposed
eigenvalue-models \cite{Dub1} with the largest k=D symmetry. Extending
\cite{Dub1}, we reconstruct a D-dimensional gauge theory with the large N free
energy given (modulo the volume factor) by the free energy of a given proposed
1\leq{k}\leq{(D-1)} D-matrix system. It is emphasized that the developed
formalism provides with the basis for higher-dimensional generalizations of the
Gross-Taylor stringy representation of strongly coupled 2d gauge theories.
