214 research outputs found
Crystal structure on rigged configurations
Rigged configurations are combinatorial objects originating from the Bethe
Ansatz, that label highest weight crystal elements. In this paper a new
unrestricted set of rigged configurations is introduced for types ADE by
constructing a crystal structure on the set of rigged configurations. In type A
an explicit characterization of unrestricted rigged configurations is provided
which leads to a new fermionic formula for unrestricted Kostka polynomials or
q-supernomial coefficients. The affine crystal structure for type A is obtained
as well.Comment: 20 pages, 1 figure, axodraw and youngtab style file necessar
Connecting marginally large tableaux and rigged configurations via crystals
We show that the bijection from rigged configurations to tensor products of
Kirillov-Reshetikhin crystals extends to a crystal isomorphism between the
models given by rigged configurations and marginally large
tableaux.Comment: 22 pages, 3 figure
Crystal structure on rigged configurations and the filling map
In this paper, we extend work of the first author on a crystal structure on
rigged configurations of simply-laced type to all non-exceptional affine types
using the technology of virtual rigged configurations and crystals. Under the
bijection between rigged configurations and tensor products of
Kirillov-Reshetikhin crystals specialized to a single tensor factor, we obtain
a new tableaux model for Kirillov-Reshetikhin crystals. This is related to the
model in terms of Kashiwara-Nakashima tableaux via a filling map, generalizing
the recently discovered filling map in type .Comment: 45 page
Rigged configurations and the Bethe Ansatz
These notes arose from three lectures presented at the Summer School on
Theoretical Physics "Symmetry and Structural Properties of Condensed Matter"
held in Myczkowce, Poland, on September 11-18, 2002. We review rigged
configurations and the Bethe Ansatz. In the first part, we focus on the
algebraic Bethe Ansatz for the spin 1/2 XXX model and explain how rigged
configurations label the solutions of the Bethe equations. This yields the
bijection between rigged configurations and crystal paths/Young tableaux of
Kerov, Kirillov and Reshetikhin. In the second part, we discuss a
generalization of this bijection for the symmetry algebra , based on
work in collaboration with Okado and Shimozono.Comment: 24 pages; lecture notes; axodraw style file require
Rigged Configurations and Kashiwara Operators
For types and we prove that the rigged configuration
bijection intertwines the classical Kashiwara operators on tensor products of
the arbitrary Kirillov-Reshetikhin crystals and the set of the rigged
configurations.Comment: v2: 108 pages, the author's final version for publication,
Proposition 33 added, Section 7.3 partially reworked; v3: published version
(Special Issue in honor of Anatol Kirillov and Tetsuji Miwa
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
Promotion operator on rigged configurations of type A
Recently, the analogue of the promotion operator on crystals of type A under
a generalization of the bijection of Kerov, Kirillov and Reshetikhin between
crystals (or Littlewood--Richardson tableaux) and rigged configurations was
proposed. In this paper, we give a proof of this conjecture. This shows in
particular that the bijection between tensor products of type A_n^{(1)}
crystals and (unrestricted) rigged configurations is an affine crystal
isomorphism.Comment: 37 page
Type rigged configuration bijection
We establish a bijection between the set of rigged configurations and the set
of tensor products of Kirillov--Reshetikhin crystals of type in
full generality. We prove the invariance of rigged configurations under the
action of the combinatorial -matrix on tensor products and show that the
bijection preserves certain statistics (cocharge and energy). As a result, we
establish the fermionic formula for type . In addition, we establish
that the bijection is a classical crystal isomorphism.Comment: 54 page
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