Relationship Between Mullineux Involution and the Generalized Regularization

The Mullineux involution is an important map on $p$-regular partitions that originates from the modular representation theory of $\mathcal{S}_n$. In this paper we study the Mullineux transpose map and the generalized column regularization and prove a condition under which the two maps are exactly the same. Our results generalize the work of Bessenrodt, Olsson and Xu, and the combinatorial constructions is related to the Iwahori-Hecke algebra and the global crystal basis of the basic $U_q(\widehat{\mathfrak{sl}}_b)$-module. In the conclusion, we provide several conjectures regarding the $q$-decomposition numbers and generalizations of results due to Fayers.