Jack polynomials with symbolic parameter

Nice achievment: I have been able to implement the Jack polynomials with a symbolic Jack parameter, in Haskell and in R. My Julia package JackPolynomials, a couple of years old, allows to get them as well:

julia> using JackPolynomials
julia> JackPolynomial(2, [3; 1])
(2*alpha^2 + 4*alpha + 2)*x_1^3*x_2 + (4*alpha + 4)*x_1^2*x_2^2 + (2*alpha^2 + 4*alpha + 2)*x_1*x_2^3

But the Haskell implementation and the R implementation required much more work. As compared to Julia, I also have to implement some stuff to deal with polynomials with symbolic coefficients. Such stuff is available in Julia, I didn’t have to implement it myself.

In Haskell, the multivariate polynomials with symbolic coefficients are implemented in my package hspray. In R, they are implemented in my package symbolicQspray (I’m tempted to rename it to parametricQspray).

Let’s compute a Jack polynomial in R:

library(jack)
JackSymPol(2, 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. The symbolicQspray package is loaded when you load the jack package. A symbolicQspray object represents a multivariate polynomial whose coefficients are fractions of polynomials with rational coefficients. However I do not consider them as polynomials on the field of fractions of polynomials, but rather as polynomials with rational coefficients depending on some parameters, the dependence being described by a fraction of polynomials. The variables of the fractions of polynomials represent these parameters.

Jack polynomials have one parameter, the Jack parameter. It is denoted by a in the above output of JackSymPol. You do not see any fraction of polynomials in this output, only polynomials. In fact, from the definition of the Jack polynomials, as well as from their implementation, the coefficients of these polynomials are fractions of polynomials in the Jack parameter. But you will never see a fraction in the output of JackSymPol: by an amazing theorem (the Knop & Sahi formula), the coefficients always are polynomials in the Jack parameter.

That explains why there are two levels of enclosing braces around the coefficients: { [ ... ] }: the right brackets [ and ] enclose the numerator of the fraction (the denominator is dropped since it is one), and the curly braces { and } enclose the fraction. I don’t want to remove the right brackets when there’s no denominator, because they indicate that the present object conceptually is a fraction of polynomials, not a polynomial. With the wording of my packages, this is a ratioOfQsprays object, not a qspray object.

The ratioOfQsprays objects are defined in my package ratioOfQsprays. A ratioOfQsprays object is nothing but a pair of qspray objects (defined in my package qspray), the numerator and the denominator, but the result of an arithmetic operation performed on these objects is always written as an irreducible fraction. This irreducible fraction is obtained thanks to the C++ library CGAL, which provides a very fast implementation of the greatest common divisor and of the division of two multivariate polynomials.

In my Haskell package hspray, the “parametric polynomials” are the objects of type SymbolicSpray (that I will possibly rename to ParametricSpray). These objects represent multivariate polynomials whose coefficients are fractions of univariate polynomials, whereas the R objects ratioOfQsprays represent fractions of multivariate polynomials. Univariate fractions of polynomials are enough for the Jack polynomials. I restricted myself to univariate polynomials because it is possible to deal with ratios of such polynomials with the numeric-prelude library, and I decided to use this stuff to have less work to achieve. But I have everything needed to introduce the fractions of multivariate polynomials and to use them as coefficients. Maybe in the future. Please let me know if you have a use case.

By the way, I’m wondering who uses my R package jack. I can see it is downloaded. Implementing the Jack polynomials was very interesting, but I do not know what they are useful for. Same question for my Python package jackpy, which has a couple of stars on Github. Note that Jack polynomials with symbolic Jack parameter are not available in this package. This should be probably doable with a moderate modification of my code.

Note that Jack polynomials are often very long. For example, but this one is not so long:

( jp <- JackSymPol(3, c(3, 1)) )
## { [ 2*a^2 + 4*a + 2 ] } * X^3.Y  +  { [ 2*a^2 + 4*a + 2 ] } * X^3.Z  +  { [ 4*a + 4 ] } * X^2.Y^2  +  { [ 6*a + 10 ] } * X^2.Y.Z  +  { [ 4*a + 4 ] } * X^2.Z^2  +  { [ 2*a^2 + 4*a + 2 ] } * X.Y^3  +  { [ 6*a + 10 ] } * X.Y^2.Z  +  { [ 6*a + 10 ] } * X.Y.Z^2  +  { [ 2*a^2 + 4*a + 2 ] } * X.Z^3  +  { [ 2*a^2 + 4*a + 2 ] } * Y^3.Z  +  { [ 4*a + 4 ] } * Y^2.Z^2  +  { [ 2*a^2 + 4*a + 2 ] } * Y.Z^3

But these polynomials are symmetric, so you can get a considerably shorter expression by writing them as a linear combination of the monomial symmetric polynomials:

compactSymmetricQspray(jp)
## [1] "{ [ 2*a^2 + 4*a + 2 ] } * M[3, 1]  +  { [ 4*a + 4 ] } * M[2, 2]  +  { [ 6*a + 10 ] } * M[2, 1, 1]"

Finally, a few words about the efficiency. Here is a benchmark in R, for the Jack polynomial of the integer partition \([4, 2, 2, 1]\) with \(5\) variables (everything is implemented in C++):

library(microbenchmark)
microbenchmark(
  JackSymPol = JackSymPol(n, lambda),
  setup = {
    n <- 5
    lambda <- c(4, 2, 2, 1)
  },
  times = 5,
  unit = "seconds"
)
## Unit: seconds
##        expr      min      lq     mean   median       uq      max neval
##  JackSymPol 2.659117 2.76353 3.041557 2.787248 3.427619 3.570271     5

Haskell is faster: it takes about \(550\) milliseconds. And unsurprisingly, Julia is the winner: about \(350\) milliseconds (with Julia 1.9.1). So the Rcpp implementation is a bit slow.