71.12 RestrictedModule

`RestrictedModule(x)`
`RestrictedModule(x, r_1 [, r_2, ...])`

Given a module x for 'H'(Sym_n) `RestrictedModule` returns the corresponding module for 'H'(Sym_{n-1}). The restriction of the Specht module `S`(mu) is the linear combination of Specht modules sum 'S'(<nu>) where nu runs over the partitions whose diagrams are obtained by deleting a node from the diagram of mu. If only nodes of residue r are deleted then this corresponds to first restricting `S`(mu) and then taking one of the block components of the restriction; this process is known as r-restriction (cf. r--induction in InducedModule).

There is also a function `SRestrictedModule` (see SRestrictedModule) which provides a faster way of r--restricting s times (and restricting s times).

When more than one residue if given to `RestrictedModule` it returns `RestrictedModule`(x,r_1,r_2,...,r_k)= `RestrictedModule`(`RestrictedModule`(x,r_1),r_2,...,r_k) (cf. `InducedModule` InducedModule).

```gap> H:=Specht(6);; RestrictedModule(H.P(5,3,2,1),4);
2*P(4,3,2,1)
gap> RestrictedModule(H.D(5,3,2),1);
D(5,2,2) ```

``Quantized'' restriction

As with `InducedModule`, when `RestrictedModule` is applied to the canonical basis elements `H.Pq`(mu) a quantum analogue of restriction is applied; this time, `RestrictedModule(*,i)` corresponds to the action of the generator E_i of U_q(widehat{sl_e}) on F [LLT].

See also `InducedModule` InducedModule, `SInducedModule` SInducedModule, and `SRestrictedModule` SRestrictedModule. This function requires the package ``specht'' (see RequirePackage).

GAP 3.4.4
April 1997