The ‘jack’ package: Jack polynomials

library(jack)

Schur polynomials have applications in combinatorics and zonal polynomials have applications in multivariate statistics. They are particular cases of Jack polynomials. This package allows to evaluate these polynomials and also to compute them in symbolic form.

Breaking change in version 6.0.0

In version 6.0.0, each function whose name ended with the suffix CPP (JackCPP, JackPolCPP, etc.) has been renamed by removing this suffix, and the functions Jack, JackPol, etc. have been renamed by adding the suffix R to their name: JackR, JackPolR, etc. The reason of these changes is that a name like Jack is more appealing than JackCPP and it is more sensible to assign the more appealing names to the functions implemented with Rcpp since they are highly more efficient. The interest of the functions JackR, JackPolR, etc. is meager.

Getting the polynomials

The functions JackPol, ZonalPol, ZonalQPol and SchurPol respectively return the Jack polynomial, the zonal polynomial, the quaternionic zonal polynomial, and the Schur polynomial.

Each of these polynomials is given by a positive integer, the number of variables (the n argument), and an integer partition (the lambda argument); the Jack polynomial has a parameter in addition, the alpha argument, a number called the Jack parameter.

To get a Jack polynomial with JackPol, you have to supply the Jack parameter as a bigq rational number or as a character string representing a fraction, e.g. "2/5":

jpol <- JackPol(2, lambda = c(3, 1), alpha = "2/5")
jpol
## 98/25*x^3.y + 28/5*x^2.y^2 + 98/25*x.y^3

This is a qspray object, from the qspray package. Here is how you can evaluate this polynomial:

evalQspray(jpol, c("2", "3/2"))
## Big Rational ('bigq') :
## [1] 1239/10

It is also possible to convert a qspray polynomial to a function whose evaluation is performed by the Ryacas package:

jyacas <- as.function(jpol)

You can provide the values of the variables of this function as numbers or character strings:

jyacas(2, "3/2")
## [1] "1239/10"

You can even pass a variable name to this function:

jyacas("x", "x")
## [1] "(336*x^4)/25"

If you want to substitute a complex number to a variable, use a character string which represents this number, with I denoting the imaginary unit:

jyacas("2 + 2*I", "2/3 + I/4")
## [1] "Complex((-158921)/2160,101689/2160)"

It is also possible to evaluate a qspray polynomial for some complex values of the variables with evalQspray. You have to separate the real parts and the imaginary parts:

evalQspray(jpol, values_re = c("2", "2/3"), values_im = c("2", "1/4"))
## Big Rational ('bigq') object of length 2:
## [1] -158921/2160 101689/2160

Direct evaluation of the polynomials

If you just have to evaluate a Jack polynomial, you don’t need to resort to a qspray polynomial: you can use the functions Jack, Zonal, ZonalQ or Schur, which directly evaluate the polynomial; this is much more efficient than computing the qspray polynomial and then applying evalQspray.

Jack(c("2", "3/2"), lambda = c(3, 1), alpha = "2/5")
## Big Rational ('bigq') :
## [1] 1239/10

However, if you have to evaluate a Jack polynomial for several values, it could be better to resort to the qspray polynomial.

Skew Schur polynomials

As of version 6.0.0, the package is able to compute the skew Schur polynomials, with the function SkewSchurPol.

Symbolic Jack parameter

As of version 6.0.0, it is possible to get a Jack polynomial with a symbolic Jack parameter in its coefficients, thanks to the symbolicQspray package.

( J <- JackSymPol(2, lambda = c(3, 1)) )
## { [ 2*a^2 + 4*a + 2 ] } * X^3.Y  +  { [ 4*a + 4 ] } * X^2.Y^2  +  { [ 2*a^2 + 4*a + 2 ] } * X.Y^3

This is a symbolicQspray object, from the symbolicQspray package.

A symbolicQspray object corresponds to a multivariate polynomial whose coefficients are fractions of polynomials with rational coefficients. The variables of these fractions of polynomials can be seen as some parameters. The Jack polynomials fit into this category: from their definition, their coefficients are fractions of polynomials in the Jack parameter. However you can see in the above output that for this example, the coefficients are polynomials in the Jack parameter (a): there’s no fraction. Actually this fact is always true for any Jack polynomial (for any Jack J-polynomial, I should say). This is an established fact and it is not obvious (it is a consequence of the Knop & Sahi formula).

You can substitute a value to the Jack parameter with the help of the substituteParameters function:

( J5 <- substituteParameters(J, 5) )
## 72*X^3.Y + 24*X^2.Y^2 + 72*X.Y^3
J5 == JackPol(2, lambda = c(3, 1), alpha = "5")
## [1] TRUE

Note that you can change the letters used to denote the variables. By default, the Jack parameter is denoted by a and the variables are denoted by X, Y, Z if there are no more than three variables, otherwise they are denoted by X1, X2, … Here is how to change these symbols:

showSymbolicQsprayOption(J, "a") <- "alpha"
showSymbolicQsprayOption(J, "X") <- "x"
J
## { [ 2*alpha^2 + 4*alpha + 2 ] } * x1^3.x2  +  { [ 4*alpha + 4 ] } * x1^2.x2^2  +  { [ 2*alpha^2 + 4*alpha + 2 ] } * x1.x2^3

If you want to have the variables denoted by x and y, do:

showSymbolicQsprayOption(J, "showMonomial") <- showMonomialXYZ(c("x", "y"))
J
## { [ 2*alpha^2 + 4*alpha + 2 ] } * x^3.y  +  { [ 4*alpha + 4 ] } * x^2.y^2  +  { [ 2*alpha^2 + 4*alpha + 2 ] } * x.y^3

Compact expression of Jack polynomials

The expression of a Jack polynomial in the canonical basis can be long. Since these polynomials are symmetric, one can get a considerably shorter expression by writing the polynomial as a linear combination of the monomial symmetric polynomials. This is what the function compactSymmetricQspray does:

( J <- JackPol(3, lambda = c(4, 3, 1), alpha = "2") )
## 3888*x^4.y^3.z + 2592*x^4.y^2.z^2 + 3888*x^4.y.z^3 + 3888*x^3.y^4.z + 4752*x^3.y^3.z^2 + 4752*x^3.y^2.z^3 + 3888*x^3.y.z^4 + 2592*x^2.y^4.z^2 + 4752*x^2.y^3.z^3 + 2592*x^2.y^2.z^4 + 3888*x.y^4.z^3 + 3888*x.y^3.z^4
compactSymmetricQspray(J) |> cat()
## 3888*M[4, 3, 1] + 2592*M[4, 2, 2] + 4752*M[3, 3, 2]

The function compactSymmetricQspray is also applicable to a symbolicQspray object, like a Jack polynomial with symbolic Jack parameter.

It is easy to figure out what is a monomial symmetric polynomial: M[i, j, k] is the sum of all monomials x^p.y^q.z^r where (p, q, r) is a permutation of (i, j, k).

The “compact expression” of a Jack polynomial with n variables does not depend on n if n >= sum(lambda):

lambda <- c(3, 1)
alpha <- "3"
J4 <- JackPol(4, lambda, alpha)
J9 <- JackPol(9, lambda, alpha)
compactSymmetricQspray(J4) |> cat()
## 32*M[3, 1] + 16*M[2, 2] + 28*M[2, 1, 1] + 24*M[1, 1, 1, 1]
compactSymmetricQspray(J9) |> cat()
## 32*M[3, 1] + 16*M[2, 2] + 28*M[2, 1, 1] + 24*M[1, 1, 1, 1]

Laplace-Beltrami operator

Just to illustrate the possibilities of the packages involved in the jack package (qspray, ratioOfQsprays, symbolicQspray), let us check that the Jack polynomials are eigenpolynomials for the Laplace-Beltrami operator on the space of homogeneous symmetric polynomials.

LaplaceBeltrami <- function(qspray, alpha) {
  n <- numberOfVariables(qspray)
  derivatives1 <- lapply(seq_len(n), function(i) {
    derivQspray(qspray, i)
  })
  derivatives2 <- lapply(seq_len(n), function(i) {
    derivQspray(derivatives1[[i]], i)
  })
  x <- lapply(seq_len(n), qlone) # x_1, x_2, ..., x_n
  # first term
  out1 <- 0L
  for(i in seq_len(n)) {
    out1 <- out1 + alpha * x[[i]]^2 * derivatives2[[i]]
  }
  # second term
  out2 <- 0L
  for(i in seq_len(n)) {
    for(j in seq_len(n)) {
      if(i != j) {
        out2 <- out2 + x[[i]]^2 * derivatives1[[i]] / (x[[i]] - x[[j]])
      }
    }
  }
  # at this step, `out2` is a `ratioOfQsprays` object, because of the divisions
  # by `x[[i]] - x[[j]]`; but actually its denominator is 1 because of some
  # simplifications and then we extract its numerator to get a `qspray` object
  out2 <- getNumerator(out2)
  out1/2 + out2
}

alpha <- "3"
J <- JackPol(4, c(2, 2), alpha)
collinearQsprays(
  qspray1 = LaplaceBeltrami(J, alpha), 
  qspray2 = J
)
## [1] TRUE