For Hecke algebras H defined over fields of characteristic zero Lascoux, Leclerc and Thibon [LLT] have described an easy, inductive, algorithm for calculating the decomposition matrices of H. Their algorithm really calculates the canonical basis, or (global) crystal basis of the Fock space; results of Grojnowski--Lusztig [Gr] show that computing this basis is equivalent to computing the decomposition matrices of H (see also [A]).
The Fock space F is an (integrable) module for the quantum group
U_q(widehat{sl}_{<e>}) of the affine special linear group. F is
a free C[v
]--module with basis the set of all Specht modules
S
(mu) for all partitions mu of all integers F =
bigoplus_nge0bigoplus_muvdash nC[v
] S
(mu); here
v
=H.info.Indeterminate
is an indeterminate over the integers (or
strictly, C). The canonical basis elements Pq
(mu) for the
U_q(widehat{sl}_e)--submodule of F generated by the
0--partition are indexed by e--regular partitions
mu. Moreover, under specialization, Pq
(mu) maps to
P
(mu). An eloquent description of the algorithm for computing
H.Pq
(mu) can be found in [LLT].
To access the elements of the Fock space Specht provides the functions:
H.Pq
(mu) qquad H.Sq
(mu)
Notice that, unlike H.P
and H.S
the only arguments which H.Pq
and H.Sq
accept are partitions. (Given that our indeterminate is v
these functions should really be called H.Pv
and H.Sv
; here
``q'' stands for ``quantum.)
The function H.Pq
computes the canonical basis element Pq
(mu)
of the Fock space corresponding to the e--regular partition mu
(there is a canonical basis for the whole of the Fock space [LT];
conjecturally, this basis can be used to compute the decomposition
matrices for the q--Schur algebra over fields of characteristic
zero). The second function returns a standard basis element
S
(mu) of F.
gap> H:=Specht(4); Specht(e=4, S(), P(), D(), Pq()) gap> H.Pq(6,2); S(6,2)+v*S(5,3) gap> RestrictedModule(last); S(6,1)+(v + v^(-1))*S(5,2)+v*S(4,3) gap> H.P(last); P(6,1)+(v + v^(-1))*P(5,2) gap> Specialized(last); P(6,1)+2*P(5,2) gap> H.Sq(5,3,2); S(5,3,2) gap> InducedModule(last,0); v^(-1)*S(5,3,3)
The modules returned by H.Pq
and H.Sq
behave very much like
elements of the Grothendieck ring of H; however, they should be
considered as elements of the Fock space. The key difference is that
when induced or restricted ``quantum'' analogues of induction and
restriction are used. These analogues correspond to the action of
U_q(widehat{sl}_{<e>}) on F [LLT].
In effect, the functions H.Pq
and H.Sq
allow computations in the
Fock space, using the functions InducedModule
InducedModule and
RestrictedModule
RestrictedModule. The functions H.S
, H.P
, and
H.D
can also be applied to elements of the Fock space, in which case
they have the expected effect. In addition, any element of the Fock
space can be specialized to give the corresponding element of the
Grothendieck ring of H
(it is because of this correspondence that we
do not make a distinction between elements of the Fock space and the
Grothendieck ring of H
).
When working over fields of characteristic zero Specht will
automatically calculate any canonical basis elements that it needs for
computations in the Grothendieck ring of H. If you are not
interested in the canonical basis elements you need never work with
them directly. If, for some reason, you do not want Specht to use
the canonical basis elements to calculate decomposition numbers then
all you need to do is Unbind
(H.Pq
).
GAP 3.4.4