Abstract
In this paper we give an alternate combinatorial description of the "$(\ell,0)$-Carter partitions''. Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas ($\textit{A q-analogue of the Jantzen-Schaper theorem}$). The condition of being an $(\ell,0)$-Carter partition is fundamentally related to the hook lengths of the partition. The representation-theoretic significance of their combinatoric on an $\ell$-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 $B(\Lambda_0)$ of the basic representation of $\widehat{\mathfrak{sl}_{\ell}}$, whose nodes are labeled by $\ell$-regular partitions. Here we give a fairly simple crystal-theoretic rule which generates all $(\ell,0)$-Carter partitions in the graph of $B(\Lambda_0)$. Dans cet article, nous donnons une description combinatoire alternative des partitions "$(\ell,0)$-Carter". Notre théorème principal est une équivalence entre notre combinatoire et celle introduite par James et Mathas ($\textit{A q-analogue of the Jantzen-Schaper theorem}$). La propriété $(\ell,0)$-Carter est fondamentalement liée aux longueurs des équerres de la partition. En terme de théorie des représentations, leur combinatoire pour une partition $\ell$-régulière permet de déterminer l'irréducibilité du module de Specht spécialisé sur l’algèbre de Hecke finie. Nous utilisons notre résultat pour déterminer leur série génératrice en fonction de la taille de la première part. Nous utilisons ensuite notre description de ces partitions au graphe cristallin $B(\Lambda _0)$ de la représentation basique de $\widehat{\mathfrak{sl}_{\ell}}$, dont les nœuds sont étiquetés par les partitions $\ell$-régulières. Nous donnons une règle cristalline relativement simple permettant d'engendrer toutes les partitions $\ell$-régulières $(\ell,0)$-Carter dans le graphe de $B(\Lambda _0)$.
Highlights
Let λ be a partition of n and l ≥ 2 be an integer
We study the generating function with the statistic being the first part of the partition
It is well known that a partition λ is an l-core if and only if all of the beads lie in the last l − 1 runners and if any bead is in the abacus, all of the numbers above it in the same runner have beads
Summary
Let λ be a partition of n and l ≥ 2 be an integer. We will use the convention (x, y) to denote the box which sits in the xth row and the yth column of the Young diagram of λ. Any partition which has no removable l-rim hooks is called an l-core. Removable l-rim hooks which are flat (i.e. those whose boxes all sit in one row) will be called horizontal l-rim hooks. An equivalent condition for the irreducibility of the Specht module indexed by an l-regular partition is that the hook lengths in a column of the partition λ are either all divisible by l or none of them are, for every column in λ (see [2] for general partitions, when l ≥ 3). All of the irreducible representations of Hn(q) have been constructed abstractly when q is a primitive lth root of unity These modules are indexed by l-regular partitions λ, and are called Dλ.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
