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.
Highlights
The Mullineux involution is a relevant map that appears in the study of the modular representations of the symmetric group and the alternating group
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
It is well known that the isomorphism classes of complex irreducible representations of the symmetric group Sn can be indexed by the set of partitions of n
Summary
It is well known that the number of isomorphism classes of irreducible representations of the symmetric group Sn over F , is equal to the number of conjugacy classes of p-regular elements of Sn ([Isa06, 15.11]), which in turn is in bijection with the p-regular partitions of n ([JK81, 6.1.2]) In this setting, understanding the tensor product with the sign representation allows to obtain a classification of irreducible F An-modules. Tensoring with the sign representation in the modular case amounts to applying m on partitions, which makes the Mullineux map a p-analogue of conjugation of partitions This way we have a classification of irreducible representations of An in characteristic p as follows.
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