294 research outputs found
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
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 . 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
Soliton cellular automaton associated with crystal base
We calculate the combinatorial matrix for all elements of
where denotes the
-perfect crystal of level , and then study the soliton cellular
automaton constructed from it. The solitons of length are identified with
elements of the -crystal . The scattering
rule for our soliton cellular automaton is identified with the combinatorial
matrix for -crystals
A crystal theoretic method for finding rigged configurations from paths
The Kerov--Kirillov--Reshetikhin (KKR) bijection gives one to one
correspondences between the set of highest paths and the set of rigged
configurations. In this paper, we give a crystal theoretic reformulation of the
KKR map from the paths to rigged configurations, using the combinatorial R and
energy functions. This formalism provides tool for analysis of the periodic
box-ball systems.Comment: 24 pages, version for publicatio
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
, 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 . In this manner we obtain generalisations of the
Perk--Schultz model.Comment: 10 pages, 2 figure
Spectrum in multi-species asymmetric simple exclusion process on a ring
The spectrum of Hamiltonian (Markov matrix) of a multi-species asymmetric
simple exclusion process on a ring is studied. The dynamical exponent
concerning the relaxation time is found to coincide with the one-species case.
It implies that the system belongs to the Kardar-Parisi-Zhang or
Edwards-Wilkinson universality classes depending on whether the hopping rate is
asymmetric or symmetric, respectively. Our derivation exploits a poset
structure of the particle sectors, leading to a new spectral duality and
inclusion relations. The Bethe ansatz integrability is also demonstrated.Comment: 46 pages, 9 figure
Noncommutative Schur polynomials and the crystal limit of the U_q sl(2)-vertex model
Starting from the Verma module of U_q sl(2) we consider the evaluation module
for affine U_q sl(2) and discuss its crystal limit (q=0). There exists an
associated integrable statistical mechanics model on a square lattice defined
in terms of vertex configurations. Its transfer matrix is the generating
function for noncommutative complete symmetric polynomials in the generators of
the affine plactic algebra, an extension of the finite plactic algebra first
discussed by Lascoux and Sch\"{u}tzenberger. The corresponding noncommutative
elementary symmetric polynomials were recently shown to be generated by the
transfer matrix of the so-called phase model discussed by Bogoliubov, Izergin
and Kitanine. Here we establish that both generating functions satisfy Baxter's
TQ-equation in the crystal limit by tying them to special U_q sl(2) solutions
of the Yang-Baxter equation. The TQ-equation amounts to the well-known
Jacobi-Trudy formula leading naturally to the definition of noncommutative
Schur polynomials. The latter can be employed to define a ring which has
applications in conformal field theory and enumerative geometry: it is
isomorphic to the fusion ring of the sl(n)_k -WZNW model whose structure
constants are the dimensions of spaces of generalized theta-functions over the
Riemann sphere with three punctures.Comment: 24 pages, 6 figures; v2: several typos fixe
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