  We investigate a 1D quantum system associated with the Ising model in a
field(the dilute $A_3$ model) by the recently developed quantum transfer matrix
(QTM) approach. A closed set of functional relations is found among variants of
fusion QTMs which are characterized by skew Young tableaux. These relations are
proved by using a quantum analogue of Jacobi-Trudi formula, together with
special features at "root of unity" . The numerical analysis on their
eigenvalues shows a remarkable coincidence with exponents characteristic to
$E_8$. From these findings, we have successfully recovered the $E_8$
Thermodynamic Bethe ansatz equation by Bazhanov et al, however, without
specific choice of strings solutions.
