Soliton cellular automaton associated with $G_2^{(1)}$G2(1) crystal base

Abstract

We calculate the combinatorial R matrix for all elements of B l ⊗ B 1 where B l denotes the G 1 -crystalB 3l . The scattering rule for our soliton cellular automaton is identified with the combinatorial R matrix for A( 1) 1 -crystals.

Soliton cellular automaton associated with $D_n^{(1)}$-crystal $B^{2,s}$

“…The proof of the following proposition is completely analogous to that of Proposition 9 in [26], except that ε i (u l ), and ϕ i (u l ) are now 0. Proposition 4.…”

“…Lemma 2 (Analogous to Lemma 4.15 in [35], Lemma 3 in [26]). Suppose there is a one-soliton state p = .…”

“…Before we prove the main result on scattering of solitons, we first prove several Lemmas relating R and T r , T ♮ . Lemma 3 (Analogous to Lemma 4.17 in [35], Lemma 4 in [26]). Assume that The proof of the following is the same as that of Lemma 5 in [26].…”

“…Lemma 3 (Analogous to Lemma 4.17 in [35], Lemma 4 in [26]). Assume that The proof of the following is the same as that of Lemma 5 in [26].…”

An Explicit Algorithm of Rigged Configuration Bijection for the Adjoint Crystal of Type $G_{2}^{(1)}$

“…The crystal structure is depicted in Figure 2.2. See also [17,18] (note the difference indexing of the Dynkin . we adopt the anti-Kashiwara convention for the tensor products of crystals as in [2].…”