$(\ell,0)$-Carter Partitions, their Crystal-Theoretic Behavior and Generating Function

Abstract

In this paper we give an alternate combinatorial description of the "$(\ell,0)$-Carter partitions". The representation-theoretic significance of these partitions is that they indicate the irreducibility of the corresponding specialized Specht module over the Hecke algebra of the symmetric group. Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas, which is in terms of hook lengths. We use our result to find a generating series which counts 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 $\widehat{{sl}_{\ell}}$, 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 $B(\Lambda_0)$.