Crystal rules for \((\ell,0)\)-JM partitions (Q1960273)

Summary: The author and \textit{M. Vazirani} [Electron. J. Comb. 15, No. 1, R130, 23 p. (2008; Zbl 1180.05123)] gave a new interpretation of what we called \(\ell \)-partitions, also known as \((\ell , 0)\)-Carter partitions. The primary interpretation of such a partition \(\lambda\) is that it corresponds to a Specht module \(S^{\lambda}\) which remains irreducible over the finite Hecke algebra \(H_n(q)\) when \(q\) is specialized to a primitive \(\ell\)th root of unity. To accomplish this we relied heavily on the description of such a partition in terms of its hook lengths, a condition provided by \textit{G. D. James} and \textit{A. Mathas} [Proc. Lond. Math. Soc., III. Ser. 74, No. 2, 241--274 (1997; Zbl 0869.20004)]. In this paper, a new description of the crystal \(reg_{\ell}\) is used which helps to extend previous results to all \((\ell , 0)\)-JM partitions (similar to \((\ell,0)\)-Carter partitions, but not necessarily \(\ell \)-regular), by using an analogous condition for hook lengths which was proven by work of \textit{S. Lyle} [J. Reine Angew. Math. 608, 93--121 (2007; Zbl 1157.20004)] and \textit{M. Fayers} [Adv. Math. 193, No. 2, 438--452 (2005; Zbl 1078.20003)].