71.10 InducedModule

InducedModule(x)
InducedModule(x, r_1 [,r_2, ...])

There is an natural embedding of 'H'(Sym_n) in 'H'(Sym_{n+1}) which in the usual way lets us define an induced 'H'(Sym_{n+1})--module for every 'H'(Sym_n)--module. The function InducedModule returns the induced modules of the Specht modules, principal indecomposable modules, and simple modules (more accurately, their image in the Grothendieck ring).

There is also a function SInducedModule (see SInducedModule) which provides a much faster way of r--inducing s times (and inducing s times).

Let mu be a partition. Then the induced module InducedModule(S(mu)) is easy to describe: it has the same composition factors as sum 'S'(<nu>) where nu runs over all partitions whose diagrams can be obtained by adding a single node to the diagram of mu.

gap> H:=Specht(2,2);
Specht(e=2, p=2, S(), P(), D())
gap> InducedModule(H.S(7,4,3,1));
S(8,4,3,1)+S(7,5,3,1)+S(7,4,4,1)+S(7,4,3,2)+S(7,4,3,1,1)
gap> InducedModule(H.P(5,3,1));
P(6,3,1)+2*P(5,4,1)+P(5,3,2)
gap> InducedModule(H.D(11,2,1));
# D(<x>), unable to rewrite <x> as a sum of simples
S(12,2,1)+S(11,3,1)+S(11,2,2)+S(11,2,1,1)

When inducing indecomposable modules and simple modules, InducedModule first rewrites these modules as a linear combination of Specht modules (using known decomposition matrices), and then induces this linear combination of Specht modules. If possible Specht then rewrites the induced module back in the original basis. Note that in the last example above, the decomposition matrix for Sym_{15} is not known by Specht; this is why InducedModule was unable to rewrite this module in the D--basis.

medskip

r--Induction

InducedModule(x, r_1 [, r_2, ...])

Two Specht modules S(mu) and S(nu) belong to the same block if and only if the corresponding partitions mu and nu have the same e--core [JM2] (see ECore). Because the e--core of a partition is determined by its (multiset of) e--residues, if S(mu) and S(nu) appear in InducedModule(S(tau)), for some partition tau, then S(mu) and S(nu) belong to the same block if and only if mu and nu can be obtained by adding a node of the same e--residue to the diagram of tau. The second form of InducedModule allows one to induce ``within blocks'' by only adding nodes of some fixed e--residue r; this is known as r-induction. Note that 0 le r.

gap> H:=Specht(4); InducedModule(H.S(5,2,1));
S(6,2,1)+S(5,3,1)+S(5,2,2)+S(5,2,1,1)
gap> InducedModule(H.S(5,2,1),0);
0*S()
gap> InducedModule(H.S(5,2,1),1);
S(6,2,1)+S(5,3,1)+S(5,2,1,1)
gap> InducedModule(H.S(5,2,1),2);
0*S()
gap> InducedModule(H.S(5,2,1),3);
S(5,2,2)

gap> EResidueDiagram(H,5,2,1);
  0  1  2  3  0
  3  0
  2

gap> H:=Specht(3);; InducedModule(H.Pq(4,2),1,2);
S(6,2)+v*S(4,4)+v^2*S(4,2,2)
gap> H.P(last);
P(6,2)