LittlewoodRichardsonRule(mu, nu)
LittlewoodRichardsonCoefficient(mu, nu, tau)
Given partitions mu of n and nu of m the module
'S'(<mu>) otimes 'S'(<nu>) is naturally an
'H'(Sym_ntimesSym_m)-module and, by inducing, we obtain an
'H'(Sym_{n+m})-module. This module has the same composition factors
as sum_nu a_munu^lambda S
(lambda), where the sum runs
over all partitions lambda of n+m and the integers
a_{munu}^lambda are the Littlewood--Richardson coefficients. The
integers a_{munu}^lambda can be calculated using a
straightforward combinatorial algorithm known as the
Littlewood--Richardson rule (see [JK]).
The function LittlewoodRichardsonRule
returns an (unordered) list of
partitions of n+m in which each partition lambda occurs
a_{munu}^lambda times. The Littlewood-Richardson coefficients are
independent of e; they can be read more easily from the computation
S(mu)*S(nu)
.
gap> H:=Specht(0);; # the generic Hecke algebra with 'R'=*C*['q'] gap> LittlewoodRichardsonRule([3,2,1],[4,2]); [ [ 4, 3, 2, 2, 1 ],[ 4, 3, 3, 1, 1 ],[ 4, 3, 3, 2 ],[ 4, 4, 2, 1, 1 ], [ 4, 4, 2, 2 ],[ 4, 4, 3, 1 ],[ 5, 2, 2, 2, 1 ],[ 5, 3, 2, 1, 1 ], [ 5, 3, 2, 2 ],[ 5, 4, 2, 1 ],[ 5, 3, 2, 1, 1 ],[ 5, 3, 3, 1 ], [ 5, 4, 1, 1, 1 ],[ 5, 4, 2, 1 ],[ 5, 5, 1, 1 ],[ 5, 3, 2, 2 ], [ 5, 3, 3, 1 ],[ 5, 4, 2, 1 ],[ 5, 4, 3 ],[ 5, 5, 2 ],[ 6, 2, 2, 1, 1], [ 6, 3, 1, 1, 1 ],[ 6, 3, 2, 1 ],[ 6, 4, 1, 1 ],[ 6, 2, 2, 2 ], [ 6, 3, 2, 1 ],[ 6, 4, 2 ],[ 6, 3, 2, 1 ],[ 6, 3, 3 ],[ 6, 4, 1, 1 ], [ 6, 4, 2 ], [ 6, 5, 1 ], [ 7, 2, 2, 1 ], [ 7, 3, 1, 1 ], [ 7, 3, 2 ], [ 7, 4, 1 ] ] gap> H.S(3,2,1)*H.S(4,2); S(7,4,1)+S(7,3,2)+S(7,3,1,1)+S(7,2,2,1)+S(6,5,1)+2*S(6,4,2)+2*S(6,4,1,1) +S(6,3,3)+3*S(6,3,2,1)+S(6,3,1,1,1)+S(6,2,2,2)+S(6,2,2,1,1)+S(5,5,2) +S(5,5,1,1)+S(5,4,3)+3*S(5,4,2,1)+S(5,4,1,1,1)+2*S(5,3,3,1)+2*S(5,3,2,2) +2*S(5,3,2,1,1)+S(5,2,2,2,1)+S(4,4,3,1)+S(4,4,2,2)+S(4,4,2,1,1)+S(4,3,3,2) +S(4,3,3,1,1)+S(4,3,2,2,1) gap> LittlewoodRichardsonCoefficient([3,2,1],[4,2],[5,4,2,1]); 3
The function LittlewoodRichardsonCoefficient
returns a single
Littlewood--Richardson coefficient (although you are really better off
asking for all of them, since they will all be calculated anyway).
See also InducedModule
InducedModule and
InverseLittlewoodRichardsonRule
InverseLittlewoodRichardsonRule.
This function requires the package ``specht'' (see
RequirePackage).
GAP 3.4.4