  The double Dyck path algebra $\mathbb{A}_{q,t}$ and its polynomial
representation first arose as a key figure in the proof of the celebrated
Shuffle Theorem of Carlsson and Mellit. A geometric formulation for an
equivalent algebra $\mathbb{B}_{q,t}$ was then given by the second author and
Carlsson and Mellit using the K-theory of parabolic flag Hilbert schemes. In
this article, we initiate the systematic study of the representation theory of
the double Dyck path algebra $\mathbb{B}_{q,t}$. We define a natural extension
of this algebra and study its calibrated representations. We show that the
polynomial representation is calibrated, and place it into a large family of
calibrated representations constructed from posets satisfying certain
conditions. We also define tensor products and duals of these representations,
thus proving (under suitable conditions) the category of calibrated
representations is generically monoidal. As an application, we prove that
tensor powers of the polynomial representation can be constructed from the
equivariant K-theory of parabolic Gieseker moduli spaces.
