The Mullineux map is an involutory bijection on the set of p-regular partitions of
any given integer n, where a partition is called p-regular if no part of it is repeated
p or more times. Many combinatorial properties of the Mullineux map make it reasonable
to view this map as a p-analogue of the transposition map T on the set of
all partitions.
Based on the work of Kleshchev [7], it has been shown
[2, 5, 14] that the Mullineux
map M has the following property.
If λ is a p-regular partition of n and Dλ
is the p-modular representation of the
symmetric group Sn labelled by λ
(see [6]) then
formula here
where sgn is the sign representation of Sn. Thus, from a representation theoretic
point of view as well, the Mullineux map is a p-analogue of the transposition map T.
The Mullineux map plays a vital role not only in the representation theory of
symmetric groups but also in other contexts. The definition of M as given in [9] is quite
complicated and to study various questions involving this map it is desirable to have
other descriptions. One alternative inductive description of M using the concept of
good boxes in a p-regular partition was given by Kleshchev [7] and this was used by
Walker to prove a result contained in Theorem 4·5 below; this work was motivated
by the investigation of Schur modules. In this paper we study a third description of
M based on the operator J on the set of p-regular
partitions defined in [13].