71.37 LittlewoodRichardsonRule

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).

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