  The dilute A$_3$ lattice model in regime 2 is in the universality class of
the Ising model in a magnetic field. Here we establish directly the existence
of an E$_8$ structure in the dilute A$_3$ model in this regime by expressing
the 1-dimensional configuration sums in terms of fermionic sums which
explicitly involve the E$_8$ root system. In the thermodynamic limit, these
polynomial identities yield a proof of the E$_8$ Rogers--Ramanujan identity
recently conjectured by Kedem {\em et al}.
  The polynomial identities also apply to regime 3, which is obtained by
transforming the modular parameter by $q\to 1/q$. In this case we find an
A$_1\times\mbox{E}_7$ structure and prove a Rogers--Ramanujan identity of
A$_1\times\mbox{E}_7$ type. Finally, in the critical $q\to 1$ limit, we give
some intriguing expressions for the number of $L$-step paths on the A$_3$
Dynkin diagram with tadpoles in terms of the E$_8$ Cartan matrix. All our
findings confirm the E$_8$ and E$_7$ structure of the dilute A$_3$ model found
recently by means of the thermodynamic Bethe Ansatz.

High Energy Physics - Theory;Exactly Solvable and Integrable Systems;Exactly Solvable and Integrable Systems