71.3 The Fock space and Hecke algebras over fields of characteristic zero

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).

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