32 research outputs found
A rigged configuration model for
We describe a combinatorial realization of the crystals and
using rigged configurations in all symmetrizable Kac-Moody types
up to certain conditions. This includes all simply-laced types and all
non-simply-laced finite and affine types
Alcove path model for
We construct a model for using the alcove path model of Lenart
and Postnikov. We show that the continuous limit of our model recovers a dual
version of the Littelmann path model for given by Li and Zhang.
Furthermore, we consider the dual version of the alcove path model and obtain
analogous results for the dual model, where the continuous limit gives the Li
and Zhang model.Comment: 19 pages, 7 figures; improvements from comments, added more figure
Rigged configurations and the -involution for generalized Kac--Moody algebras
We construct a uniform model for highest weight crystals and for
generalized Kac--Moody algebras using rigged configurations. We also show an
explicit description of the -involution on rigged configurations for
: that the -involution interchanges the rigging and the
corigging. We do this by giving a recognition theorem for using the
-involution. As a consequence, we also characterize as a
subcrystal of using the -involution. We show that the
category of highest weight crystals for generalized Kac--Moody algebras is a
coboundary category by extending the definition of the crystal commutor using
the -involution due to Kamnitzer and Tingley.Comment: 23 pages, 1 figur
Crystal bases and q-identities
The relation of crystal bases with -identities is discussed, and some new
results on crystals and -identities associated with the affine Lie algebra
are presented.Comment: 25 pages, style file axodraw.sty require
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
A bijection between type D_n^{(1)} crystals and rigged configurations
Hatayama et al. conjectured fermionic formulas associated with tensor
products of U'_q(g)-crystals B^{r,s}. The crystals B^{r,s} correspond to the
Kirillov--Reshetikhin modules which are certain finite dimensional
U'_q(g)-modules. In this paper we present a combinatorial description of the
affine crystals B^{r,1} of type D_n^{(1)}. A statistic preserving bijection
between crystal paths for these crystals and rigged configurations is given,
thereby proving the fermionic formula in this case. This bijection reflects two
different methods to solve lattice models in statistical mechanics: the
corner-transfer-matrix method and the Bethe Ansatz.Comment: 38 pages; version to appear in J. Algebr
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