jack (version 6.1.0)

LRmult: Littlewood-Richardson rule for multiplication

Description

Expression of the product of two Schur polynomials as a linear combination of Schur polynomials.

Usage

LRmult(mu, nu, output = "dataframe")

Value

This computes the expression of the product of the two Schur polynomials associated to mu and nu as a linear combination of Schur polynomials. If output="dataframe", the output is a dataframe with two columns: the column coeff gives the coefficients of this linear combination, these are positive integers, and the column

lambda gives the partitions defining the Schur polynomials of this linear combination as character strings, e.g. the partition

c(4, 3, 1) is encoded by the character string "[4, 3, 1]". If output="list", the output is a list of lists with two elements. Each of these lists with two elements corresponds to a term of the linear combination: the first element, named coeff, is the coefficient, namely the Littlewood-Richardson coefficient \(c^{\lambda}_{\mu,\nu}\), where \(\lambda\) is the integer partition given in the second element of the list, named

lambda, which defines the Schur polynomial of the linear combination.

Arguments

mu, nu

integer partitions, given as vectors of decreasing integers

output

the type of the output, "dataframe" or "list"

Examples

library(jack)
mu <- c(2, 1)
nu <- c(3, 2, 1)
LR <- LRmult(mu, nu, output = "list")
LRterms <- lapply(LR, function(lr) {
  lr[["coeff"]] * SchurPol(3, lr[["lambda"]])
})
smu_times_snu <- Reduce(`+`, LRterms)
smu_times_snu == SchurPol(3, mu) * SchurPol(3, nu) # should be TRUE