Affine crystal structure on rigged configurations of type D_n^(1)

Extending the work arXiv:math/0508107, we introduce the affine crystal action on rigged configurations which is isomorphic to the Kirillov-Reshetikhin crystal B^{r,s} of type D_n^(1) for any r,s. We also introduce a representation of B^{r,s} (r not equal to n-1,n) in terms of tableaux of rectangular shape r x s, which we coin Kirillov-Reshetikhin tableaux (using a non-trivial analogue of the type A column splitting procedure) to construct a bijection between elements of a tensor product of Kirillov-Reshetikhin crystals and rigged configurations.Comment: 26 pages, 3 figures. (v3) corrections in the proof reading. (v2) 26 pages; examples added; introduction revised; final version. (v1) 24 page

Energy Functions in Box Ball Systems

The box ball system is studied in the crystal theory formulation. New conserved quantities and the phase shift of the soliton scattering are obtained by considering the energy function (or $H$-function) in the combinatorial $R$-matrix.Comment: 15 pages, LaTeX2e: one paragraph replaced and reference added in Introduction, a paragraph added in Section 2.5, remark 2) after Th 4.6 adde

Crystals for Demazure Modules of Classical Affine Lie Algebras

We study, in the path realization, crystals for Demazure modules of affine Lie algebras of types $A^{(1)}_n,B^{(1)}_n,C^{(1)}_n,D^{(1)}_n, A^{(2)}_{2n-1},A^{(2)}_{2n}, and D^{(2)}_{n+1}$. We find a special sequence of affine Weyl group elements for the selected perfect crystal, and show if the highest weight is l\La_0, the Demazure crystal has a remarkably simple structure.Comment: Latex, 28 page

Generalised Perk--Schultz models: solutions of the Yang-Baxter equation associated with quantised orthosymplectic superalgebras

The Perk--Schultz model may be expressed in terms of the solution of the Yang--Baxter equation associated with the fundamental representation of the untwisted affine extension of the general linear quantum superalgebra $U_q[sl(m|n)]$, with a multiparametric co-product action as given by Reshetikhin. Here we present analogous explicit expressions for solutions of the Yang-Baxter equation associated with the fundamental representations of the twisted and untwisted affine extensions of the orthosymplectic quantum superalgebras $U_q[osp(m|n)]$. In this manner we obtain generalisations of the Perk--Schultz model.Comment: 10 pages, 2 figure

Integrable Structure of Multispecies Zero Range Process

We present a brief review on integrability of multispecies zero range process in one dimension introduced recently. The topics range over stochastic R matrices of quantum affine algebra Uq(An⁽¹⁾), matrix product construction of stationary states for periodic systems, q-boson representation of Zamolodchikov-Faddeev algebra, etc. We also introduce new commuting Markov transfer matrices having a mixed boundary condition and prove the factorization of a family of R matrices associated with the tetrahedron equation and generalized quantum groups at a special point of the spectral parameter