Mullineux(e|H, mu)
Mullineux(d, mu)
Mullineux(x)
Given an integer e, or a Specht record H, and a partition
mu, Mullineux(e, mu) returns the image of mu under
the Mullineux map; which we now explain.
The sign representation D(1^n) of the Hecke algebra is the (one
dimensional) representation sending T_w to (-1)^{ell(w)}. The
Hecke algebra H is not a Hopf algebra so there is no well defined
action of H upon the tensor product of two H--modules; however,
there is an outer automorphism # of H which corresponds to
tensoring with D(1^n). This sends an irreducible module
'D'(<mu>) to an irreducible 'D'(<mu>)^#cong 'D'(<mu^#>) for
some e--regular partition mu^#. In the symmetric group case,
Mullineux gave a conjectural algorithm for calculating mu^#;
consequently the map sending mu to mu^# is known as the
Mullineux map.
Deep results of Kleshchev [K] for the symmetric group give another (proven) algorithm for calculating the partition mu^# (Ford and Kleshchev have deduced Mullineux's conjecture from this). Using the canonical basis, it was shown by [LLT] that the natural generalization of Kleshchev's algorithm to H gives the Mullineux map for Hecke algebras over fields of characteristic zero. The general case follows from this, so the Mullineux map is now known for all Hecke algebras.
Kleshchev's map is easy to describe; he proved that if gns is any
good node sequence for mu, then the sequence obtained from gns
by replacing each residue r by -rbmod e is a good node sequence
for mu^# (see GoodNodeSequence GoodNodeSequence).
gap> Mullineux(Specht(2),12,5,2); [ 12, 5, 2 ] gap> Mullineux(Specht(4),12,5,2); [ 4, 4, 4, 2, 2, 1, 1, 1 ] gap> Mullineux(Specht(6),12,5,2); [ 4, 3, 2, 2, 2, 2, 2, 1, 1 ] gap> Mullineux(Specht(8),12,5,2); [ 3, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 ] gap> Mullineux(Specht(10),12,5,2); [ 3, 3, 3, 3, 2, 1, 1, 1, 1, 1 ]
Mullineux(d, mu)
The Mullineux map can also be calculated using a decomposition matrix.
To see this recall that ``tensoring'' a Specht module S(mu)
with the sign representation yields a module isomorphic to the dual of
S(lambda), where lambda is the partition conjugate to
mu. It follows that d_{munu}=d_{lambdanu^#} for all
e--regular partitions nu. Therefore, if mu is the last
partition in the lexicographic order such that d_{munu}ne0 then
we must have nu^#=lambda. The second form of Mullineux uses d
to calculate mu^# rather than the Kleshchev--[LLT] result.
Mullineux(x)
In the third form, x is a module, and Mullineux returns <x>^#,
the image of x under #. Note that the above remarks show that
P(mu) is mapped to P(mu^#) via the Mullineux map; this
observation is useful when calculating decomposition matrices (and is
used by the function InducedDecompositionMatrix).
See also GoodNodes GoodNodes and GoodNodeSequence
GoodNodeSequence . This function requires the package ``specht''
(see RequirePackage).
GAP 3.4.4