This is the first in a series of papers in which we describe explicit structural properties of spaces of diagonal rectangular harmonic polynomials in
k
k
sets of
n
n
variables, both as
G
L
k
GL_k
-modules and
S
n
\mathbb {S}_n
-modules, as well as some of there relations to areas such as Algebraic Combinatorics, Representation Theory, Algebraic Geometry, Knot Theory, and Theoretical Physics. Our global aim is to develop a unifying point of view for several areas of research of the last two decades having to do with Macdonald Polynomials Operator Theory, Diagonal Coinvariant Spaces, Rectangular-Catalan Combinatorics, the Delta-Conjecture, Hilbert Scheme of Points in the Plane, Khovanov-Rozansky Homology of
(
m
,
n
)
(m,n)
-Torus links, etc.