scholarly journals Demazure Crystals, Kirillov-Reshetikhin Crystals, and the Energy Function

10.37236/2184 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Anne Schilling ◽  
Peter Tingley
Keyword(s):  
Lie Algebras ◽  
Energy Function ◽  
Tensor Products ◽  
Macdonald Polynomials ◽  
Close Connection ◽  
Whittaker Functions ◽  
Demazure Character ◽  
Demazure Crystals

It has previously been shown that, at least for non-exceptional Kac-Moody Lie algebras, there is a close connection between Demazure crystals and tensor products of Kirillov-Reshetikhin crystals. In particular, certain Demazure crystals are isomorphic as classical crystals to tensor products of  Kirillov-Reshetikhin crystals via a canonically chosen isomorphism. Here we show that this isomorphism intertwines the natural affine grading on Demazure crystals with a combinatorially defined energy function. As a consequence, we obtain a formula of the Demazure character in terms of the energy function, which has applications to Macdonald polynomials and $q$-deformed Whittaker functions.

scholarly journals Demazure crystals and the energy function

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Anne Schilling ◽  
Peter Tingley
Keyword(s):  
Energy Function ◽  
Tensor Products ◽  
Macdonald Polynomials ◽  
Close Connection ◽  
Whittaker Functions ◽  
Nous Obtenons ◽  
Nous Montrons ◽  
International Audience ◽  
Demazure Character ◽  
Demazure Crystals

International audience There is a close connection between Demazure crystals and tensor products of Kirillov–Reshetikhin crystals. For example, certain Demazure crystals are isomorphic as classical crystals to tensor products of Kirillov–Reshetikhin crystals via a canonically chosen isomorphism. Here we show that this isomorphism intertwines the natural affine grading on Demazure crystals with a combinatorially defined energy function. As a consequence, we obtain a formula of the Demazure character in terms of the energy function, which has applications to nonsymmetric Macdonald polynomials and $q$-deformed Whittaker functions. Les cristaux de Demazure et les produits tensoriels de cristaux Kirillov–Reshetikhin sont étroitement liés. Par exemple, certains cristaux de Demazure sont isomorphes, en tant que cristaux classiques, à des produits tensoriels de cristaux Kirillov–Reshetikhin via un isomorphisme que l'on peut choisir canoniquement. Ici, nous montrons que cet isomorphisme entremêle la graduation affine naturelle des cristaux de Demazure avec une fonction énergie définie combinatoirement. Comme conséquence, nous obtenons une formule pour le caractère de Demazure exprimée au moyen de la fonction énergie, avec des applications aux polynômes de Macdonald non symétriques et aux fonctions de Whittaker $q$-déformées.

scholarly journals Crystal energy via charge

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Cristian Lenart ◽  
Anne Schilling
Keyword(s):  
Energy Function ◽  
Tensor Products ◽  
Macdonald Polynomials ◽  
Recursive Definition ◽  
R Matrix ◽  
Single Column ◽  
Nous Montrons ◽  
International Audience ◽  
Definition Of

International audience The Ram–Yip formula for Macdonald polynomials (at t=0) provides a statistic which we call charge. In types ${A}$ and ${C}$ it can be defined on tensor products of Kashiwara–Nakashima single column crystals. In this paper we show that the charge is equal to the (negative of the) energy function on affine crystals. The algorithm for computing charge is much simpler than the recursive definition of energy in terms of the combinatorial ${R}$-matrix. La formule de Ram et Yip pour les polynômes de Macdonald (à t = 0) fournit une statistique que nous appelons la charge. Dans les types ${A}$ et ${C}$, elle peut être définie sur les produits tensoriels des cristaux pour les colonnes de Kashiwara–Nakashima. Dans ce papier, nous montrons que la charge est égale à (l'opposé de) la fonction d'énergie sur cristaux affines. L'algorithme pour calculer la charge est bien plus simple que la définition récursive de l'énergie en fonction de la ${R}$-matrice combinatoire.

scholarly journals A generalization of the alcove model and its applications

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Cristian Lenart ◽  
Arthur Lubovsky
Keyword(s):  
Lie Algebras ◽  
Energy Function ◽  
Tensor Products ◽  
Present Evidence ◽  
Semisimple Lie Algebras ◽  
Highest Weight ◽  
International Audience ◽  
Alcove Model

International audience The alcove model of the first author and Postnikov describes highest weight crystals of semisimple Lie algebras. We present a generalization, called the quantum alcove model, and conjecture that it uniformly describes tensor products of column shape Kirillov-Reshetikhin crystals, for all untwisted affine types. We prove the conjecture in types $A$ and $C$. We also present evidence for the fact that a related statistic computes the energy function. Le modèle des alcôves du premier auteur et Postnikov décrit les cristaux de plus haut poids des algèbres de Lie semi-simples. Nous présentons une généralisation, appelée le modèle des alcôves quantique, et nous conjecturons qu’il décrit dans une manière uniforme les produits tensoriels des cristaux de Kirillov-Reshetikhin de type colonne, pour toutes les types affines symétriques. Nous prouvons la conjecture dans les types $A$ et $C$. Nous fournissons aussi des preuves qu’une statistique associée donne la fonction d’énergie.

scholarly journals Lakshmibai–Seshadri paths of level-zero shape and one-dimensional sums associated to level-zero fundamental representations

2008 ◽  
Vol 144 (6) ◽  
pp. 1525-1556 ◽  
Author(s):  
Satoshi Naito ◽  
Daisuke Sagaki
Keyword(s):  
Energy Function ◽  
Tensor Products ◽  
Quantum Affine Algebras ◽  
One Dimensional ◽  
Level Zero ◽  
Affine Algebras

AbstractWe give an interpretation of the energy function and classically restricted one-dimensional sums associated to tensor products of level-zero fundamental representations of quantum affine algebras in terms of Lakshmibai–Seshadri paths of level-zero shape.

NEW EXCEPTIONAL $(A_1^{(1)})^{\oplus r} $ INVARIANTS AND THE ASSOCIATED GEPNER MODELS

1992 ◽  
Vol 07 (10) ◽  
pp. 2245-2264 ◽  
Author(s):  
JÜRGEN FUCHS ◽  
ALBRECHT KLEMM ◽  
MICHAEL G. SCHMIDT ◽  
DIRK VERSTEGEN
Keyword(s):  
Lie Algebras ◽  
Infinite Series ◽  
Tensor Products ◽  
Heterotic String ◽  
Partition Functions ◽  
Affine Lie Algebras ◽  
Affine Algebra ◽  
Modular Invariant ◽  
Fusion Rules ◽  
Modular Invariants

New modular invariant partition functions for tensor products of [Formula: see text] affine Lie algebras are presented. These exceptional modular invariants can be understood in terms of automorphisms of the fusion rules of the affine algebra or of its extensions. There are three isolated cases, as well as an infinite series of new invariants. As an application, the new modular invariants are employed to produce new Gepner type compactifications of the heterotic string.

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