  We construct a new family of graded representations $\widetilde{W}_{\lambda}$
for the positive elliptic Hall algebra $\mathcal{E}^{+}$ indexed by Young
diagrams $\lambda$ which generalize the standard $\mathcal{E}^{+}$ action on
symmetric functions. These representations have homogeneous bases of
eigenvectors for the action of the Macdonald element $P_{0,1} \in
\mathcal{E}^{+}$ with distinct $\mathbb{Q}(q,t)$-rational spectrum generalizing
the symmetric Macdonald functions. The analysis of the structure of these
representations exhibits interesting combinatorics arising from the stable
limits of periodic standard Young tableaux. We find an explicit combinatorial
rule for the action of the multiplication operators $e_r[X]^{\bullet}$
generalizing the Pieri rule for symmetric Macdonald functions. We will also
naturally obtain a family of interesting $(q,t)$ product-series identities
which come from keeping track of certain combinatorial statistics associated to
periodic standard Young tableaux.
