The AM(1) automata related to crystals of symmetric tensors

Abstract

A soliton cellular automaton associated with crystals of symmetric tensor representations of the quantum affine algebraIt is a crystal theoretic formulation of the generalized box-ball system in which capacities of boxes and carriers are arbitrary and inhomogeneous. Scattering matrices of two solitons coincide with the combinatorial R matrices of U ′ q (AM −1 ). A piecewise linear evolution equation of the automaton is identified with an ultradiscrete limit of the nonautonomous discrete KP equation. A class of N soliton solutions is obtained through the ultradiscretization of soliton solutions of the latter.(1) M ) corresponding to the k-fold symmetric tensor representation. As a set it consists of the single row semistandard tableaux of length k on letters {1, 2, . . . , M + 1}:where we have omitted the k − 1 vertical lines separating the entries. We also represent the elements by the multiplicities of their contents. Namely, b = m 1 · · · m k ∈ B k is also denoted by b = (x 1 , x 2 , · · · , x M +1 ) with x i = #{l | m l = i}.

A crystal theoretic method for finding rigged configurations from paths

“…Remark 2.2 Piecewise linear formula to obtain the combinatorial R and the energy function is also available [25]. This is suitable for computer implementation.…”

Kirillov-Schilling-Shimozono Bijection as Energy Functions of Crystals

“…We remark that this isomorphism is nothing but time evolution of the box-ball systems [3,2]. Then our main result (Theorem 3.3) states that by taking differences of these E l or e i 's, we can derive alternative algorithm of the map φ.…”

Untitled

“…The diagram should not be confused with the one representing the combinatorial R matrix. 4 Given i, (b ′ , s ′ ) is uniquely fixed from (b, s). Thus for example the diagram…”

“…Their integrability has been understood by the ultradiscretization 28 of classical integrable systems (soliton equations). In the recent works 6,3,4 it was revealed that the box-ball systems may also be viewed as quantum integrable systems at q = 0. Here by quantum integrable systems we mean the ones whose integrability is guaranteed by the Yang-Baxter equation, 1 and q is the deformation parameter in the relevant quantum group.…”