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

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

2011-07-19

arxiv.org/abs/1107.3613v1

Electronic Journal of Combinatorics, Volume 17 (1), 2010

Mathematics

Combinatorics

Scientific paper

Vazirani and the author \cite{BV} 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 James and Mathas. In this paper, I use a new description of the crystal $reg_\ell$ which helps 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 Lyle and Fayers.

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