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).

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GAP 3.4.4
April 1997