(l, 0)-Carter Partitions, their Crystal-Theoretic Behavior and Generating Function

In this paperwe give an alternate combinatorial description of the “(l, 0)-Carter partitions” (see [4]). 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 (see [7]). Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas ([7]), 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 graphB(Λ0) of the basic representation of ŝll, 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 B(Λ0).