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
The function
medskip
``Quantized'' induction
When
See also
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 rgap> 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)
EResidueDiagram
(EResidueDiagram), prints the diagram
of mu, labeling each node with its e--residue. A quick check of
this diagram confirms the answers above.
gap> EResidueDiagram(H,5,2,1);
0 1 2 3 0
3 0
2
InducedModule
is applied to the canonical basis elements
H.Pq
(mu) (or more generally elements of the Fock space; see
Specht), a ``quantum analogue'' of induction is applied. More
precisely, the function InducedModule(*,i)
corresponds to the
action of the generator F_i of the quantum group
U_q(widehat{sl_e}) on F [LLT].
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)
SInducedModule
SInducedModule, RestrictedModule
RestrictedModule, and SRestrictedModule
SRestrictedModule. This
function requires the package ``specht'' (see RequirePackage).
April 1997