($\ell,0)$-Carter partitions, a generating function, and their crystal theoretic interpretation

TL;DR

This paper provides a new combinatorial description of $( ext{ell},0)$-JM partitions, relates it to crystal graphs, and uses it to count such partitions and analyze their representation-theoretic significance.

Contribution

It introduces an alternative combinatorial characterization of $( ext{ell},0)$-JM partitions and connects this to crystal graph structures and representation theory.

Findings

Established equivalence with James and Mathas's characterization.

Derived a generating function for counting partitions by first part.

Provided a crystal-theoretic rule for generating all relevant partitions.

Abstract

In this paper we give an alternate combinatorial description of the "$(\ell,0)$-JM partitions" (see \cite{F}) that are also $\ell$-regular. Our main theorem is the equivalence of our combinatoric and the one introduced by James and Mathas (\cite{JM}). The condition of being an $(\ell,0)$-JM 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 specialized Specht module over the finite Hecke algebra (see \cite{JM}). 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 $\hat{\mathfrak{sl}_{\ell}}$, whose nodes are labeled by $\ell$-regular partitions. Here we give a fairly simple crystal-theoretic rule which generates all $\ell$-regular $(\ell,0)$-JM partitions in the graph $B(\Lambda_0)$. Finally, we mention how our construction can be generalized to recent results of M. Fayers (see \cite{F}) and S. Lyle (see \cite{L}) to count the total number of (not necessarily $\ell$-regular) Specht modules which stay irreducible at a primitive $\ell$th root of unity (for $\ell >2$).