A solvable vertex model in ferromagnetic regime gives rise to a soliton cellular automaton which is a discrete dynamical system in which site variables take on values in a finite set. We study the scattering of a class of soliton cellular automata associated with the

$U_q(D_n^{(1)})$
Uq(Dn(1))-perfect crystal B2, s. We calculate the combinatorial R matrix for all elements of B2, sB2, 1. In particular, we show that the scattering rule for our soliton cellular automaton can be identified with the combinatorial R matrix for
$U_q(A_1^{(1)}) \oplus U_q(D_{n-2}^{(1)})$
Uq(A1(1))Uq(Dn2(1))
-crystals.

1.
Drinfel'd
,
V.
, “
Hopf algebras and the quantum Yang-Baxter equation
,”
Sov. Math. Dokl.
32
(
1
),
254
258
(
1985
).
2.
Fourier
,
G.
,
Okado
,
M.
, and
Schilling
,
A.
, “
Kirillov-Reshetikhin crystals for nonexceptional types
,”
Adv. Math.
222
(
3
),
1080
1116
(
2009
).
3.
Fourier
,
G.
,
Okado
,
M.
, and
Schilling
,
A.
, “
Perfectness of Kirillov-Reshetikhin crystals for nonexceptional types
,”
Contemp. Math.
506
,
127
143
(
2010
).
4.
Fukuda
,
K.
,
Yamada
,
Y.
, and
Okado
,
M.
, “
Energy functions in box-ball systems
,”
Int. J. Mod. Phys. A
15
(
9
),
1379
1392
(
2000
).
5.
Hatayama
,
G.
,
Hikami
,
K.
,
Inoue
,
R.
,
Kuniba
,
A.
,
Takagi
,
T.
, and
Tokihiro
,
T.
, “
The
$A_M^{(1)}$
AM(1)
automata related to crystals of symmetric tensors
,”
J. Math. Phys.
42
(
3
),
274
308
(
2001
).
6.
Hatayama
,
G.
,
Kuniba
,
A.
, and
Takagi
,
T.
, “
Soliton cellular automata associated with crystal bases
,”
Nucl. Phys. B
577
(
3
),
619
645
(
2000
).
7.
Hatayama
,
G.
,
Kuniba
,
A.
,
Okado
,
M.
, and
Takagi
,
T.
, “
Combinatorial R matrices for a family of crystals:
$B^{(1)}_n, D^{(1)}_n, A^{(2)}_{2n}$
Bn(1),Dn(1),A2n(2)
, and
$D^{(2)}_{n + 1}$
Dn+1(2)
cases
,”
J. Algebra
247
(
2
),
577
615
(
2002
).
8.
Hatayama
,
G.
,
Kirillov
,
A.
,
Kuniba
,
A.
,
Okado
,
M.
,
Takagi
,
T.
, and
Yamada
,
Y.
, “
Character formulae of
$\widehat{sl}_n$
sl̂n
-modules and inhomogeneous paths
,”
Nucl. Phys. B
536
,
575
616
(
1998
).
9.
Hatayama
,
G.
,
Kuniba
,
A.
,
Okado
,
M.
,
Takagi
,
T.
, and
Yamada
,
Y.
, “
Remarks on fermionic formula
,”
Contemp. Math.
248
,
243
291
(
1999
).
10.
Hatayama
,
G.
,
Kuniba
,
A.
,
Okado
,
M.
,
Takagi
,
T.
, and
Yamada
,
Y.
, “
Scattering rules on soliton cellular automata associated with crystal bases
,”
Contemp. Math.
297
,
151
182
(
2002
).
11.
Hong
,
J.
and
Kang
,
S.-J.
,
Introduction to Quantum Groups and Crystal Bases
,
Graduate Studies in Mathematics
Vol.
42
(
American Mathematical Society
,
Providence, RI
,
2002
).
12.
Inoue
,
R.
,
Kuniba
,
A.
, and
Takagi
,
T.
, “
Integrable structure of box-ball systems:crystal, Bethe ansatz, ultradiscretization and tropical geometry
,”
J. Phys. A: Math. Theor.
45
,
073001
(
2012
).
13.
Jimbo
,
M.
, “
A q-difference analog of
$U(\mathfrak {g})$
U(g)
and the Yang-Baxter equation
,”
Lett. Math. Phys.
10
(
1
),
63
69
(
1985
).
14.
Kac
,
V. G.
,
Infinite Dimensional Lie Algebras
, 3rd ed. (
Cambridge University Press
,
Cambridge, England
,
1990
).
15.
Kang
,
S.-J.
,
Kashiwara
,
M.
, and
Misra
,
K. C.
, “
Crystal bases of Verma modules for quantum affine Lie algebras
,”
Compos. Math.
92
,
299
325
(
1994
); https://eudml.org/doc/90309.
16.
Kang
,
S.-J.
,
Kashiwara
,
M.
,
Misra
,
K. C.
,
Miwa
,
T.
,
Nakashima
,
T.
, and
Nakayashiki
,
A.
, “
Affine crystals and vertex models
,”
Int. J. Mod. Phys. A
7
(
suppl. 1A
),
449
484
(
1992
).
17.
Kashiwara
,
M.
, “
Crystallizing the q-analog of universal enveloping algebras
,”
Commun. Math. Phys.
133
,
249
260
(
1990
).
18.
Kashiwara
,
M.
,
Misra
,
K. C.
,
Okado
,
M.
, and
Yamada
,
D.
, “
Perfect crystals of
$U_q(D_4^{(3)})$
Uq(D4(3))
,”
J. Algebra
317
,
392
423
(
2007
).
19.
Kashiwara
,
M.
and
Nakashima
,
T.
, “
Crystal graphs for representations of the q-analogs of classical Lie algebras
,”
J. Algebra
165
,
295
345
(
1994
).
20.
Kirillov
,
A.
and
Reshetikhin
,
N.
, “
Representations of Yangians and multiplicity of occurrence of the irreducible components of the tensor product of representations of simple Lie algebras
,”
J. Sov. Math.
52
,
3156
3164
(
1990
).
21.
Kuniba
,
A.
,
Takagi
,
T.
, and
Takenouchi
,
A.
, “
Factorization, reduction and embedding in integrable cellular automata
,”
J. Phys. A
37
(
3
),
1691
1709
(
2004
).
22.
Lecouvey
,
C.
, “
Combinatorics of crystal graphs for the root systems of types An, Bn, Cn, Dn, and G2
,”
MSJ Memoirs
(
Mathematical Society of Japan
,
Tokyo
,
2007
), Vol.
17
, pp.
11
41
.
23.
Lusztig
,
G.
, “
Canonical bases arising from quantized universal enveloping algebras
,”
J. Amer. Math. Soc.
3
(
2
),
447
498
(
1990
).
24.
Lusztig
,
G.
,
Introduction to Quantum Groups
,
Progress in Mathematics
Vol.
110
(
Birkhäuser
,
Boston
,
1993
).
25.
Mohamad
,
M.
, “
Scattering rules in soliton cellular automata associated with
$U^{\prime }_q(D_n^{(1)})$
Uq(Dn(1))
-crystal Bn, l
,”
J. Phys. A
45
(
7
),
075208
(
2012
).
26.
Misra
,
K. C.
,
Mohamad
,
M.
, and
Okado
,
M.
, “
Zero action on perfect crystals for
$U_q(G_2^{(1)})$
Uq(G2(1))
,”
SIGMA
201
,
022
(
2010
), http://www.doaj.org/doaj?func=openurl&genre=article&issn=18150659&date=2010&volume=6&issue=&spage=022.
27.
Misra
,
K. C.
,
Okado
,
M.
, and
Wilson
,
E. A.
, “
Soliton cellular automaton associated with
$G_2^{(1)}$
G2(1)
-crystal base
,”
J. Math. Phys.
53
,
013510
(
2012
).
28.
Okado
,
M.
and
Schilling
,
A.
, “
Existence of Kirillov-Reshetikhin crystals for nonexceptional types
,”
Representation Theory
12
,
186
207
(
2008
).
29.
Schilling
,
A.
, “
Combinatorial structure of Kirillov-Reshetikhin crystals of type
$D^{(1)}_n, B^{(1)}_n, A^{(2)}_{2n+1}$
Dn(1),Bn(1),A2n+1(2)
,”
J. Algebra
319
(
7
),
2938
2962
(
2008
).
30.
Schilling
,
A.
and
Sternberg
,
P.
, “
Finite-dimensional crystals B2, s for quantum affine algebras of type
$D_n^{(1)}$
Dn(1)
,”
J. Algebr. Comb.
23
,
317
354
(
2006
).
31.
Shimozono
,
M.
, “
Affine type A crystal structure on tensor products of rectangles, Demazure characters, and nilpotent varieties
,”
J. Algebr. Comb.
15
,
151
187
(
2002
).
32.
Takahashi
,
D.
, “
On some soliton systems defined by using boxes and balls
,” in
Proceedings of the International Symposium on Nonlinear Theory and Its Applications (NOLTA '93)
,
1993
, pp.
555
558
.
33.
Takahashi
,
D.
and
Satsuma
,
J.
, “
A soliton cellular automaton
,”
J. Phys. Soc. Jpn.
59
,
3514
3519
(
1990
).
34.
Tokihiro
,
T.
,
Nagai
,
A.
, and
Satsuma
,
J.
, “
Proof of the solitonical nature of box and ball systems by means of inverse ultra-discretization
,”
Inverse Probl.
15
,
1639
1662
(
1999
).
35.
Tokihiro
,
T.
,
Takahashi
,
D.
, and
Matsukidaira
,
J.
, “
Box and ball system as a realization of ultadiscrete nonautonomous KP equation
,”
J. Phys. A.
33
,
607
619
(
2000
).
36.
Yamada
,
D.
, “
Box-ball system associated with antisymmetric tensor crystals
,”
J. Phys. A.
37
(
42
),
9975
9987
(
2004
).
37.
Yamada
,
D.
, “
Scattering rule in soliton cellular automata associated with crystal base of
$U_q(D_4^{(3)})$
Uq(D4(3))
,”
J. Math. Phys.
48
,
043509
(
2007
).
You do not currently have access to this content.