Affine structures and a tableau model for E_6 crystals

Abstract

We provide the unique affine crystal structure for type E_6^{(1)} Kirillov-Reshetikhin crystals corresponding to the multiples of fundamental weights s Lambda_1, s Lambda_2, and s Lambda_6 for all s \geq 1 (in Bourbaki's labeling of the Dynkin nodes, where 2 is the adjoint node). Our methods introduce a generalized tableaux model for classical highest weight crystals of type E and use the order three automorphism of the affine E_6^{(1)} Dynkin diagram. In addition, we provide a conjecture for the affine crystal structure of type E_7^{(1)} Kirillov-Reshetikhin crystals corresponding to the adjoint node.

28 pages

References in corpus (5)

• Existence of Kirillov-Reshetikhin crystals for nonexceptional types
• Perfect Crystals for U_q(D_4^{(3)})
• Combinatorial structure of Kirillov-Reshetikhin crystals of type D_n(1), B_n(1), A_{2n-1}(2)
• Full heaps and representations of affine Weyl groups
• A generalization of adjoint crystals for the quantized affine algebras of type $A\sb{n}\sp{(1)}$, $C\sb{n}\sp{(1)}$ and $D\sb{n+1}\sp{(2)}$

Cited by in corpus (15)

• Path description of type B q-characters
• Crystal energy functions via the charge in types A and C
• A crystal to rigged configuration bijection and the filling map for type $D_4^{(3)}$
• On higher level Kirillov--Reshetikhin crystals, Demazure crystals, and related uniform models
• KKR type bijection for the exceptional affine algebra E_6^{(1)}
• Poset models for Weyl group analogs of symmetric functions and Schur functions
• A uniform approach to soliton cellular automata using rigged configurations
• Virtualization map for the Littelmann path model
• Uniform description of the rigged configuration bijection
• A combinatorial realization of Kirillov-Reshetikhin crystals for type E arising from translations
• Kirillov-Reshetikhin crystals $B^{7,s}$ for type $E_7^{(1)}$
• Kirillov-Reshetikhin crystals $B^{1,s}$ for ${\widehat{\mathfrak{sl}}}_n$ using Nakajima monomials
• Quantum loop algebras and l-root operators
• Demazure crystals, Kirillov-Reshetikhin crystals, and the energy function
• Perfectness of Kirillov-Reshetikhin Crystals $B^{r,s}$ for types $E_6^{(1)}$ and $E_7^{(1)}$ with a minuscule node $r$