LRmult function

Littlewood-Richardson rule for multiplication

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

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

Arguments

• mu, nu: integer partitions, given as vectors of decreasing integers
• output: the type of the output, "dataframe" or "list"

Returns

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.

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