Rigged configuration descriptions of the crystals B(∞) and B(λ) for special linear Lie algebras

@article{Hong2017RiggedCD,
  title={Rigged configuration descriptions of the crystals B(∞) and B($\lambda$) for special linear Lie algebras},
  author={Jin Hong and Hyeonmi Lee},
  journal={Journal of Mathematical Physics},
  year={2017},
  volume={58},
  pages={101701}
}
The rigged configuration realization RC(∞) of the crystal B(∞) was originally presented as a certain connected component within a larger crystal. In this work, we make the realization more concrete by identifying the elements of RC(∞) explicitly for the An-type case. Two separate descriptions of RC(∞) are obtained. These lead naturally to isomorphisms RC(∞)≅T(∞) and RC(∞)≅T¯(∞), i.e., those with the marginally large tableau and marginally large reverse tableau realizations of B(∞), that may be… 
2 Citations
Characterization of ${\cal B}(\infty)$ using marginally large tableaux and rigged configurations in the $A_n$ case via integer sequences
Rigged configurations are combinatorial objects prominent in the study of solvable lattice models. Marginally large tableaux are semi-standard Young tableaux of special form that give a realization
A Complete Bibliography of Publications in the Journal of Mathematical Physics: 2005{2009
(2 < p < 4) [200]. (Uq(∫u(1, 1)), oq1/2(2n)) [92]. 1 [273, 79, 304, 119]. 1 + 1 [252]. 2 [352, 318, 226, 40, 233, 157, 299, 60]. 2× 2 [185]. 3 [456, 363, 58, 18, 351]. ∗ [238]. 2 [277]. 3 [350]. p

References

SHOWING 1-10 OF 47 REFERENCES
Crystal B(λ) as a subset of crystal B(∞) expressed as tableaux for An type
Affine crystal structure on rigged configurations of type $D_{n}^{(1)}$
Extending the work in Schilling (Int. Math. Res. Not. 2006:97376, 2006), we introduce the affine crystal action on rigged configurations which is isomorphic to the Kirillov–Reshetikhin crystal Br,s
A rigged configuration model for B(∞)
Crystal Bases and Young Tableaux
Let B be the crystal basis of the minus part of the quantized enveloping algebra of a semi-simple Lie algebra. Kashiwara has shown that B has a combinatorial description in terms of an embedding of B
On crystal bases of the $Q$-analogue of universal enveloping algebras
0. Introduction. The notion of the q-analogue of universal enveloping algebras is introduced independently by V. G. Drinfeld and M. Jimbo in 1985 in their study of exactly solvable models in the
Canonical bases for the quantum group of type $A_r$ and piecewise-linear combinatorics
This work was motivated by the following two problems from the classical representation theory. (Both problems make sense for an arbitrary complex semisimple Lie algebra but since we shall deal only
Paths, crystals and fermionic formulae
We introduce a fermionic formula associated with any quantum affine algebra U q (X N (r) . Guided by the interplay between corner transfer matrix and the Bethe ansatz in solvable lattice models, we
String bases for quantum groups of type ᵣ
This is the quantum deformation (or q−deformation) of the algebra of polynomial functions on the group Nr+1 of upper unitriangular (r + 1) × (r + 1) matrices. In this paper we introduce and study a
Polyhedral Realizations of Crystal Bases for Quantized Kac-Moody Algebras
Let B(\infty) be the crystal corresponding to the nilpotent part of a quantized Kac-Moody algebra. We suggest a general way to represent B(\infty) as the set of integer solutions of a system of
...
...