Virtual crystals and Kleber's algorithm

Abstract

Kirillov and Reshetikhin conjectured what is now known as the fermionic formula for the decomposition of tensor products of certain finite dimensional modules over quantum affine algebras. This formula can also be extended to the case of $q$-deformations of tensor product multiplicities as recently conjectured by Hatayama et al. (math.QA/9812022 and math.QA/0102113). In its original formulation it is difficult to compute the fermionic formula efficiently. Kleber (q-alg/9611032 and math.QA/9809087) found an algorithm for the simply-laced algebras which overcomes this problem. We present a method which reduces all other cases to the simply-laced case using embeddings of affine algebras. This is the fermionic analogue of the virtual crystal construction by the authors, which is the realization of crystal graphs for arbitrary quantum affine algebras in terms of those of simply-laced type.

23 pages; style file youngtab.sty required, package pstricks required; fixed typo in Eq. (5.2)

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• Categorical relations between Langlands dual quantum affine algebras: Exceptional cases
• Finite-Dimensional Crystals B^{2,s} for Quantum Affine Algebras of type D_{n}^{(1)}
• A rigged configuration model for $B(\infty )$
• Crystal structure on rigged configurations
• Crystal energy functions via the charge in types A and C
• Rigged configuration bijection and proof of the $X=M$ conjecture for nonexceptional affine types
• A crystal to rigged configuration bijection and the filling map for type $D_4^{(3)}$
• A bijection between type D_n^{(1)} crystals and rigged configurations
• Type $D_n^{(1)}$ rigged configuration bijection
• On the X=M=K Conjecture
• On higher level Kirillov--Reshetikhin crystals, Demazure crystals, and related uniform models
• Rigged configurations for all symmetrizable types
• Crystal structure on rigged configurations and the filling map
• X=M Theorem: Fermionic formulas and rigged configurations under review
• Rigged configurations and the $\ast$-involution
• Connecting marginally large tableaux and rigged configurations via crystals
• Virtualization map for the Littelmann path model
• Identities from representation theory
• Existence of Kirillov--Reshetikhin crystals for multiplicity free nodes
• A uniform approach to soliton cellular automata using rigged configurations
• Uniform description of the rigged configuration bijection
• Applications of Crystal Bases to Current Problems in Representation Theory
• Demazure crystals, Kirillov-Reshetikhin crystals, and the energy function
• Virtual Crystals and Nakajima Monomials
• An Explicit Algorithm of Rigged Configuration Bijection for the Adjoint Crystal of Type $G_2^{(1)}$
• Rigged Configuration Descriptions of the Crystals B(infinity) and B(lambda) for Special Linear Lie Algebras