DMTCS Proceedings, 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)

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(ℓ,0)-Carter Partitions and their crystal theoretic interpretation

Chris Berg, Monica Vazirani


In this paper we give an alternate combinatorial description of the ``(ℓ,0)-Carter partitions''. Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas (A q-analogue of the Jantzen-Schaper theorem). The condition of being an (ℓ,0)-Carter partition is fundamentally related to the hook lengths of the partition. The representation-theoretic significance of their combinatoric on an ℓ-regular partition is that it indicates the irreducibility of the corresponding Specht module over the finite Hecke algebra. We use our result to find a generating series which counts the number of such partitions, with respect to the statistic of a partition's first part. We then apply our description of these partitions to the crystal graph B(Λ0) of the basic representation of &widehat;slℓ, whose nodes are labeled by ℓ-regular partitions. Here we give a fairly simple crystal-theoretic rule which generates all (ℓ,0)-Carter partitions in the graph of B(Λ0).
Résumé. Dans cet article, nous donnons une description combinatoire alternative des partitions «(ℓ,0)-Carter». Notre theoreme principal est une equivalence entre notre combinatoire et celle introduite par James et Mathas (A q-analogue of the Jantzen-Schaper theorem). La propriete (ℓ,0)-Carter est fondamentalement liee aux longueurs des equerres de la partition. En terme de theorie des representations, leur combinatoire pour une partition ℓ-reguliere permet de determiner l'irreducibilite du module de Specht specialise sur l'algebre de Hecke finie. Nous utilisons notre resultat pour determiner leur serie generatrice en fonction de la taille de la premiere part. Nous utilisons ensuite notre description de ces partitions au graphe cristallin B(Λ0) de la representation basique de &widehat;slℓ, dont les noeuds sont etiquetes par les partitions ℓ-regulieres. Nous donnons un regle cristalline relativement simple permettant d'engendrer toutes les partitions ℓ-regulieres (ℓ,0)-Carter dans le graphe de B(Λ0).

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