In part 1 we study a class of cellular automata associated with the Kirillov-Reshetikhin
crystal B
n;1
of type D
(1)
n
. They have a commuting family of time evolutions and
solitons of length l are labeled by U
q
(A
(1)
n1
)-crystal B
2;l
A
. The scattering rule of two
solitons of lengths l
1
and l
2
(l
1
> l
) including the phase shift is identified with the
combinatorial R-matrix for the U
q
2
(A
(1)
n1
)-crystal B
2;l
A
2
B
2;l
A
1
. In part 2 we consider
the Kirrilov-Reshetikhin crystal B
6;1
for the exceptional affine type E
. We will
give a conjecture on a statistic-preserving bijection between the highest weight paths
consisting of B
6;1
and the corresponding rigged configuration. The algorithm only uses
the structure of the crystal graph, hence could also be applied for other exceptional
types. Our B
6;1
has a different algorithm compared our B
1;1
because we must consider
the element Ø, unique element in the highest weight crystal of weight 0, in the crystal
graph. We will give many examples supporting the conjecture.
(1)
