On self-Mullineux and self-conjugate partitions

TL;DR

This paper explores the fixed points of the Mullineux involution in modular representation theory, establishing an explicit bijection with self-conjugate partitions using the Mullineux symbol.

Contribution

It provides a new explicit bijection between self-Mullineux partitions and self-conjugate partitions, enhancing understanding of their combinatorial relationship.

Findings

Established a bijection between self-Mullineux and self-conjugate partitions

Connected fixed points of Mullineux involution with self-conjugate partitions

Utilized the Mullineux symbol to describe the bijection

Abstract

The Mullineux involution is a relevant map that appears in the study of the modular representations of the symmetric group and the alternating group. The fixed points of this map are certain partitions of particular interest. It is known that the cardinality of the set of these self-Mullineux partitions is equal to the cardinality of a distinguished subset of self-conjugate partitions. In this work, we give an explicit bijection between the two families of partitions in terms of the Mullineux symbol.